Category Archives: History of Astronomy

The black sheep of the Provence-Paris group

I continue my sketches of the seventeenth century group pf mathematicians and astronomers associated with Nicolas-Claude Fabri de Peiresc (1580-1637) in Provence and Marin Mersenne (1588–1648) in Paris with Jean-Baptiste Morin (1583–1656), who was born in Villefranche-sur-Saône in the east of France.

Jean-Baptiste Morin Source: Wikimedia Commons

He seems to have come from an affluent family and at the age of sixteen he began his studies at the University of Aix-en-Provence. Here he resided in the house of the Provencal astronomer Joseph Gaultier de la Valette (1564–1647), vicar general of Aix and Peiresc’s observing partner. For the last two years of his time in Aix, the young Pierre Gassendi, also lived in Gaultier de la Valette’s house and the two became good friends and observing partners.

In 1611, Morin moved to the University of Avignon, where he studied medicine graduating MD in 1613. For the next eight years, until 1621, he was in the service of Claude Dormy (c.1562–1626) the Bishop of Boulogne, in Paris, who paid for him to travel extensively in Germany, Hungary and Transylvania to study the metal mining industry. As well as serving Dormy as physician, he almost certainly acted as his astrologer, this was still in the period when astro-medicine or iatromathematics was the mainstream medical theory.

The tomb of Claude Dormy Source

From 1621 to 1629 he served Philip IV, King of Spain, and Duke of Luxembourg, also probably as astrologer. 

In 1630, he was indirectly asked by Marie de’ Medici, the Queen Mother, to cast a horoscope for her son, Louis XIII, who was seriously ill and whose doctor had predicted, on his own astrological reading, that he would die. Morin’s astrological analysis said that Louis would be severely ill but would survive. Luckily for Morin, his prediction proved accurate, and Marie de’ Medici used her influence to have him appointed professor for mathematics at the Collège Royal in Paris, a position he held until his death in 1656.

Marie de Médici portrait by Frans Pourbus, the Younger Source: Wikimedia Commons

In Paris, Morin he took up his friendship with Gassendi from their mutual student days and even continued to make astronomical observations with him in the 1630s, at the same time becoming a member of the group around Mersenne. However, in my title I have labelled Morin the black sheep of the Provence-Paris group and if we turn to his scholarly activities, it is very clear why. Whereas Peiresc, Mersenne, Boulliau, and Gassendi were all to one degree or another supporters of the new scientific developments in the early seventeenth century, coming to reject Aristotelean philosophy and geocentric astronomy in favour of a heliocentric world view, Morin stayed staunchly conservative in his philosophy and his cosmology.

Already in 1624, Morin wrote and published a defence of Aristotle, and he remained an Aristotelian all of his life. He rejected heliocentricity and insisted that the Earth lies at the centre of the cosmos and does not move. Whereas the others in the group supported the ideas of Galileo and also tried to defend Galileo against the Catholic Church, Morin launched an open attack on Galileo and his ideas in 1630, continuing to attack him even after his trial and house arrest. In 1638, he also publicly attacked René Descartes and his philosophy, not critically like Gassendi, but across-the-board, without real justification. He famously wrote that he knew that Descartes philosophy was no good just by looking at him when they first met. This claim is typical of Morin’s character, he could, without prejudice, be best described as a belligerent malcontent. Over the years he managed to alienate himself from almost the entire Parisian scholarly community. 

It would seem legitimate to ask, if Morin was so pig-headed and completely out of step with the developments and advances in science that were going on around him, and in which his friends were actively engaged, why bother with him at all? Morin distinguished himself in two areas, one scientific the other pseudo-scientific and it is to these that we now turn.

The scientific area in which made a mark was the determination of longitude. With European seamen venturing out into the deep sea for the first time, beginning at the end of the fifteenth century, navigation took on a new importance. If you are out in the middle of one of the Earth’s oceans, then being able to determine your exact position is an important necessity. Determining one’s latitude is a comparatively easy task. You need to determine local time, the position of the Sun, during the day, or the Pole Star, during the night and then make a comparatively easy trigonometrical calculation. Longitude is a much more difficult problem that relies on some method of determining time differences between one’s given position and some other fixed position. If one is one hour time difference west of Greenwich, say, then one is fifteen degrees of longitude west of Greenwich. 

Finding a solution to this problem became an urgent task for all of the European sea going nations, including France, and several of them were offering substantial financial rewards for a usable solution. In 1634, Morin suggested a solution using the Moon as a clock. The method, called the lunar distance method or simply lunars, was not new and had already suggested by the Nürnberger mathematicus, Johannes Werner (1468–1522) in his Latin translation of Ptolemaeus’ GeographiaIn Hoc Opere Haec Continentur Nova Translatio Primi Libri Geographicae Cl Ptolomaei, published in Nürnberg in 1514 and then discussed by Peter Apian (1495–1552) in his Cosmographicus liber, published in Landshut in 1524.

The lunar distance method relies on determining the position of the Moon relative to a given set of reference stars, a unique constellation for every part of the Moon’s orbit. Then using a set of tables to determine the timing of a given constellation for a given fixed point. Having determined one’s local time, it is then possible to calculate the time difference and thus the longitude. Because it is pulled hither and thither by both the Sun and the Earth the Moon’s orbit is extremely erratic and not the smooth ellipse suggested by Kepler’s three laws of planetary motion. This led to the realisation that compiling the tables to the necessary accuracy was beyond the capabilities of those sixteenth century astronomers and their comparatively primitive instruments, hence the method had not been realised. Another method that was under discussion was taking time with you in the form of an accurate clock, as first proposed by Gemma Frisius (1508–1555), Morin did not think much of this idea:

“I do not know if the Devil will succeed in making a longitude timekeeper but it is folly for man to try.”

Morin was well aware of the difficulties involved and suggested a comprehensive plan to overcome them. Eager to win the offered reward money, Morin put his proposal to Cardinal Richelieu (1585–1642), Chief Minister and most powerful man in France. Morin suggested improved astronomical instruments fitted out with vernier scales, a recent invention, and telescopic sights, also comparatively new, along with improvements in spherical trigonometry. He also suggested the construction of a national observatory, with the specific assignment of collected more accurate lunar data. Richelieu put Morin’s proposition to an expert commission consisting of Étienne Pascal (1588–1651), the father of Blaise, Pierre Hérigone (1580–1643), a Parisian mathematics teacher, and Claude Mydorge (1585–1647), optical physicist and geometer. This commission rejected Morin’s proposal as still not practical, resulting in a five year long dispute between Morin and the commission. It would be another century before Tobias Mayer (1723–1762) made the lunar distance method viable, basically following Morin’s plan.

Although his proposal was rejected, Morin did receive 2000 livre for his suggestion from Richelieu’s successor, Cardinal Mazarin (1602–1661) in 1645. Mazarin’s successor Jean-Baptiste Colbert (1619–1683) set up both the Académie des sciences in 1666 and the Paris Observatory in 1667, to work on the problem. This led, eventually to Charles II setting up the Royal Observatory in Greenwich, in 1675 for the same purpose.

Today, Morin is actually best known as an astrologer. The practice of astrology was still acceptable for mathematicians and astronomers during Morin’s lifetime, although it went into steep decline in the decades following his death. Although an avid astronomer, Peiresc appears to have had no interest in astrology. This is most obvious in his observation notes on the great comet of 1618. Comets were a central theme for astrologers, but Peiresc offers no astrological interpretation of the comet at all. Both Mersenne and Gassendi accepted the scientific status of astrology and make brief references to it in their published works, but neither of them appears to have practiced astrology. Boulliau also appear to have accepted astrology, as amongst his published translations of scientific texts from antiquity we can find Marcus Manilius’ Astronomicom (1655), an astrological poem written about 30 CE, and Ptolemaeus’ De judicandi jacultate (1667). Like Mersenne and Gassendi he appears not to have practiced astrology.

According to Morin’s own account, he initially thought little of astrology, but around the age of thirty he changed his mind and then spent ten years studying it in depth.

Jean-Baptiste Morin’s with chart as cast by himself

He then spent thirty years writing a total of twenty-six volumes on astrology that were published posthumously as one volume of 850 pages in Den Hague in 1661, as Astrologia Galllica (French Astrology). Like Regiomontanus, Tycho Brahe, and Kepler before him, he saw astrology as in need of reformation and himself as its anointed reformer. 

Source: Wikimedia Commons

The first eight volumes of Astrologia Galllica hardly deal with astrology at all but lay down Morin’s philosophical and religious views on which he bases his astrology. The remaining eighteen volumes then deal with the various topics of astrology one by one. Central to his work is the concept of directio in interpreting horoscopes. This is a method of determining the times of major events in a subject’s life that are indicated in their birth horoscope. Originally, to be found in Ptolemaeus’ Tetrabiblos, it became very popular during the Renaissance. The standard text was Regiomontanus’ Tabulae Directionum, originally written in 1467, and large numbers of manuscripts can still be found in libraries and archives. It was published in print by Erhard Ratdolt in Augsburg in 1490 and went through eleven editions, the last being published in 1626. Aware of Kepler’s rejection of both the signs of the zodiac and the system of houses, Morin defends both of them.

Coming, as it did at a time when astrology was in decline as an accepted academic discipline, Morin’s Astrologia Galllica had very little impact in the seventeenth century, but surprisingly, in English translation, it enjoys a lot of popularity amongst modern astrologers.

Morin was cantankerous and belligerent, which cost him most of his contacts with the contemporary scholars in Paris and his adherence to Aristotelian philosophy and a geocentric world view put him out of step with the rest of the Provence-Paris group, but he certainly didn’t suffer from a lack of belief in his own abilities as he tells us in this autobiographical quote:

“… I am excessively inclined to consider myself superior to others on account of my intellectual endowments and scientific attainments, and it is very difficult for me to struggle against this tendency, except when the realization of my sins troubles me, and I see myself a vile man and worthy of contempt. Because of all this my name has become famous throughout the world.”


1 Comment

Filed under History of Astrology, History of Astronomy, History of medicine, History of Navigation


I really shouldn’t but the HISTSCI_HULK is twisting my arm and muttering dark threats, so here goes. A week ago, we took apart Vedang Sati’s post 10 Discoveries By Newton That Changed The World. When I copied it to my blog, I removed the links that Sati had built into his post. I then made the mistake of following his link to his post on Kepler, so here we go again. 

Johannes Kepler Source: Wikimedia Commons

7 Ways In Which Johannes Kepler Changed Astronomy

Johannes Kepler was a German astronomer who discovered the three laws of planetary motion. Apart from his contributions to astronomy, he is also known to have pioneered the field of optics. In this post, let’s read some amazing facts about Kepler and his work. 

He obviously doesn’t rate Kepler as highly as he rates Newton, so the introduction is less hagiographic this time. However, it does contain one quite extraordinary claim, when he writes, “he is also known to have pioneered the field of optics.” Optics as a scientific discipline was pioneered by Euclid, who lived in the fourth century BCE, so about two thousand years before Kepler. There were also quite a few people active in the field in the two millennia in between.

Early Affliction

He suffered from small pox at a very early age. The disease left him with weak eyesight. Isn’t  it wonderful then how he went on to invent eyeglasses for near-eye and far-eye sightedness.

Kepler did indeed suffer from smallpox sometime around the age of four, which almost cost him his life and did indeed leave him with damaged eyesight. However, Kepler did not invent spectacles of any type whatsoever. The first spectacles for presbyopia, far-sightedness occurring in old age, began to appear in the last decades of the thirteenth century CE. Spectacles for myopia, short-sightedness, were widely available by the early fifteenth century. What Kepler actually did was to publish the first scientific explanation of how lenses function to correct defects in eyesight in his Astronomiae Pars Optica (The Optical Part of Astronomy), in 1604. Francesco Maurolico (1494–1574) actually gave the correct explanation earlier than Kepler in his Photismi de lumine et umbra but this was only published posthumously in 1611, so the credit for priority goes to Kepler

Astronomiae Pars Optica Source: Wikimedia Commons

Introduction to Astronomy

Kepler’s childhood was worsened by his family’s financial troubles. At the age of 6, Johannes had to drop out of school so to earn money for the family. He worked as a waiter in an inn.

As Kepler first entered school at the age of seven, it would have been difficult for his schooling to have been interrupted when he was six. His primary schooling was in fact often interrupted both by illness and the financial fortunes of the family. 

In the same year, his mother took him out at night to show him the Great Comet of 1577 which aroused his life-long interest in science and astronomy. 

Yes, she did!

Copernican Supporter

At a time when everyone was against the heliocentric model of the universe, Kepler became its outspoken supporter. He was the first person to defend the Copernican theory from a scientific and a religious perspective.

Not everyone was opposed to the heliocentric model of the universe, just the majority. Poor old Georg Joachim Rheticus (1514–1574), as the professor of mathematics, who persuaded Copernicus to publish De revolutionibus, he would be deeply insulted by the claim that Kepler was the “first person to defend the Copernican theory from a scientific and a religious perspective.” Rheticus, of course, did both, long before Kepler was even born, although his religious defence remained unpublished and was only rediscovered in the twentieth century. Rheticus was not the only supporter of Copernicus, who preceded Kepler there were others, most notably, in this case, Michael Mästlin (1550–1631), who taught Kepler the Copernican heliocentrism. 

Contemporary of Galileo

Galileo was not a great supporter of Kepler’s work especially when Kepler had proposed that the Moon had an influence over the water (tides). It would take an understanding many decades later which would prove Kepler correct and Galileo wrong.

It is indeed very true that Galileo rejected Kepler’s theory of the tides, when promoting his own highly defective theory, but that is mild compared to his conscious ignoring of Kepler’s laws of planetary motion, which were at the time the most significant evidence for a heliocentric cosmos.

Pioneer of Optics

Kepler made ground-breaking contributions to optics including the formulation of inverse-square law governing the intensity of light; inventing an improved refracting telescope; and correctly explaining the function of the human eye.

Kepler’s contributions to the science of optics were indeed highly significant and represent a major turning point in the development of the discipline. His Astronomiae Pars Optica does indeed contain the inverse square law of light intensity and the first statement that the image is created in the eye on the retina and not in the crystal lens.

However, that he invented an improved telescope is more than a little problematic. When Galileo published his Sidereus Nuncius in 1610, the first published account of astronomical, telescopic discoveries, there was no explanation how a telescope actually functions, so people were justifiably sceptical. Having written the book on how lenses function with his Astronomiae Pars Optica in 1604, Kepler now delivered a scientific explanation how the telescope functioned with his Dioptrice in 1611. 

Kepler Dioptrice Source: Wikimedia Commons

This contained not just a theoretical explanation of the optics of a Dutch or Galilean telescope, with a convex objective and a concave eyepiece, but also of a telescope with convex objective and convex eyepiece, which produces an inverted image, now known as a Keplerian or astronomical telescope, also one with three convex lenses, the third lens to right the inverted image, now known as a field telescope, and lastly, difficult to believe, the telephoto lens. Kepler’s work remained strictly theoretical, and he never constructed any of these telescopes, so is he really the inventor? The first astronomical telescope was constructed by Christoph Grienberger (1561–1636) for Christoph Scheiner (c. 1573–1650) as his heliotropic telescope, for his sunspot studies. 

Heliotropic telescope on the left. On the right Scheiner’s acknowledgement that Grienberger was the inventor

Is the astronomical telescope an improved telescope, in comparison with the Dutch telescope? It is very much a question of horses for courses. If you go to the theatre or the opera then your opera glasses, actually a Dutch telescope, will be much more help in distinguishing the figure on the stage than an astronomical telescope. Naturally, the astronomical telescope, with its wider fields of vision, is, as its name implies, much better for astronomical observations than the Dutch telescope once you get past the problem of the inverted image. This problem was solved with the invention of the multiple lens eyepiece by Anton Maria Schyrleus de Rheita (1604–1660), announced in Oculus Enoch et Eliae published in 1645, although he had already been manufacturing them together with Johann Wiesel (1583–1662) since 1643.

Helped Newton

His planetary laws went on to help Sir Isaac Newton derive the inverse square law of gravity. Newton had famously acknowledged Kepler’s role, in a quote: “If I have seen further, it is by standing on the shoulders of giant(s).

Sati is not alone in failing to give credit to Kepler for his laws of planetary motion in their own right, but instead regarding them merely as a stepping-stone for Newton and the law of gravity. Kepler’s laws of planetary motion, in particular his third law, are the most significant evidence for a heliocentric model of the cosmos between the publication of De revolutionibus in 1543 and Principia in 1687 and deserve to be acknowledged and honoured in their own right! 

Newton’s famous quote, actually a much-used phrase in one form or another in the Early Modern period, originated with Bernard of Chartres (died after 1124) in the twelfth century. Newton used it in a letter to Robert Hooke on 5 February 1675, so twelve years before the publication of his Principia and definitively not referencing Kepler:

What Des-Cartes [sic] did was a good step. You have added much several ways, & especially in taking the colours of thin plates into philosophical consideration. If I have seen further it is by standing on the sholders [sic] of Giants.

Kepler’s Legacy

There is a mountain range in New Zealand named after the famous astronomer. A crater on the Moon is called Kepler’s crater. NASA paid tribute to the scientist by naming their exo-planet telescope, Kepler.

Given the vast number of things named after Kepler, particularly in Germany, Sati’s selection is rather bizarre, in particular because it is a mountain hiking trail in New Zealand that is named after Kepler and not the mountain range itself.

Once again, we are confronted with a collection of half facts and straight falsehoods on this website, whose author, as I stated last time has nearly 190,000 followers on Facebook. 

Me: I told you that we couldn’t stop the tide coming in

HS_H: You’re not trying hard enough. You’ve gotta really STOMP EM!

Me: #histsigh


Filed under History of Astronomy, History of Optics, Myths of Science

Renaissance science – XXVI

I wrote a whole fifty-two-part blog post series on The Emergence of Modern Astronomy, much of which covered the same period as this series, so I’m not going to repeat it here. However, an interesting question is, did the developments in astronomy during the Humanist Renaissance go hand in hand with humanism and to what extent, or did the two movements run parallel in time to each other without significant interaction? 

The simple answer to my own questions is yes, humanism and the emergence of modern astronomy were very closely interlinked in the period between 1400 and the early seventeenth century. This runs contrary to a popular conception that the Humanist Renaissance was purely literary and in no way scientific. In what follows I will briefly sketch some of that interlinking. 

To start, two of the driving forces behind the desire to renew and improve astronomy, the rediscovery of Ptolemaic mathematics-based cartography and the rise in importance of astrology were very much part of the Humanist Renaissance, as I have already documented in earlier episodes of this series. It is not a coincidence that many of the leading figures in the development of modern astronomy were also involved, either directly or indirectly, in the new cartography. Also, nearly all of them were active astrologers. 

Turning to the individual astronomers, the man, who kicked off the debate on the astronomical status of comets, a debate that, I have shown, played a central role in the evolution of modern astronomy, Paolo dal Pozzo Toscanelli (1397–1482) a member of the Florentine circle of prominent humanist scholars that included Filippo Brunelleschi, Marsilio Ficino, Leon Battista Alberti and Cardinal Nicolaus Cusanus, all of whom have featured in earlier episodes of this series.

Paolo dal Pozzo Toscanelli Source: Wikimedia Commons

Toscanelli, who is best known as the cosmographer, who supplied Columbus with a misleading world map, was one of those who met the Neoplatonic philosopher Georgius Gemistus Pletho (c. 1355–c. 1452) at the Council of Florence. Pletho introduced Toscanelli to the work of the Greek geographer Strabo (c. 64 BCE–c. 24 CE), which was previously unknown in Italy. 

Turning to the University of Vienna and the so-called First Viennese School of Mathematics, already during the time of Johannes von Gmunden (c. 1380–1442) and Georg Müstinger (before 1400–1442), Vienna had become a major centre for the new cartography as well as astronomy. However, it is with the next generation that we find humanist scholars at work. Toscanelli’s unpublished work on comets might have remained unknown if it hadn’t been for Georg von Peuerbach (1423–1461). As a young man Peuerbach had travelled extensively in Italy and become acquainted with the circle of humanists to which Toscanelli belonged. He shared an apartment in Rome with Cusanus and personally met and exchanged ideas with Toscanelli. Returning to Vienna he lectured on poetics and took a leading role in reviving classical Greek and Latin literature, a central humanist concern. Today he is, of course, better known for his work as an astronomer and as the teacher of Johannes Müller, better known Regiomontanus.

First page of Peuerbach’s Theoricae novae planetarum in the Manuscript Krakau, Biblioteca Jagiellońska, Ms. 599, fol. 1r (15th century) Source: Wikimedia Commons

Regiomontanus (1436–1476) became a member of the familia (household) of the leading Greek humanist scholar Basilios Bessarion (1403–1472), a pupil of Pletho. He travelled with Bessarion through Italy, working as his librarian finding and copying Latin and Greek manuscripts on astronomy, astrology and mathematics for Bessarion’s library. Bessarion had taught him Greek for this purpose. Leaving Bessarion’s service Regiomontanus served as librarian for the humanist scholars, János Vitéz Archbishop of Esztergom (c. 1408–1472) a friend of Peuerbach’s and then Matthias Corvinus (1443–1490) King of Hungary. 

Regiomontanus woodcut from the 1493 Nuremberg Chronicle Source: Wikimedia Commons

When Regiomontanus left Hungary for Nürnberg he took a vast collection of Geek and Latin manuscripts with him, with the intention of printing them and publishing them. At the same time applying humanist methods of philology to free them of their errors accumulated through centuries of copying and recopying. A standard humanist project as was the Epitome of Ptolemaeus that he and Peuerbach produced under the stewardship of Bessarion.

The so-called Second Viennese School of mathematics was literally founded by a humanist, when Conrad Celtis (1459–1508) took the professors of mathematics Andreas Stiborius (1464–1515) and Johann Stabius (before 1468–1522), along with the student Georg Tanstetter (1482–1535) from Ingolstadt to Vienna, where he founded his Collegium poetarum et mathematicorum, that is a college for poetry and mathematics, in 1497. Ingolstadt had established the first ever German chair for mathematics to teach astrology to medical students, also basically a humanist driven development.

Conrad Celtis: In memoriam by Hans Burgkmair the Elder, 1507
Source: Wikimedia Commons

The wind of humanism was strong in Vienna, where Peter Apian (1495–1552) was Tanstetter’s star pupil becoming like his teacher a cosmographer, returning to Ingolstadt, where his star pupil was his own son Philipp (1531–1589), like his father a cosmographer. Philipp became professor in Tübingen, where he was Michael Mästlin’s teacher, instilling him with the Viennese humanism. As should be well known Mästlin was Kepler’s teacher.

Source: Wikimedia Commons

Back-tracking, we must consider the central figure of the emergence of modern astronomy, Nicolaus Copernicus (1473–1543). There are no doubts about Copernicus’ humanist credentials.

Copernicus holding lily-of-the-valley: portrait in Nicolaus Reusner’s Icones (1587) Source: Wikimedia Commons

He initially studied at the University of Krakow, the oldest humanist university in Europe north of the Italian border. He continued his education at various North Italian humanist universities, where he continued to learn his astronomy from the works of Peuerbach and Regiomontanus (as he had already done in Krakow) under the supervision of Domenico Maria da Novara (1454–1504) a Neoplatonist, who regarded himself as a student of Regiomontanus.

Domenico Maria da Novara Source Museo Galileo

In Northern Italy Copernicus received a full humanist education even learning Greek and some Hebrew. Establishing his humanist credentials, Copernicus published a Latin translation from the Greek of a set of 85 brief poems by the seventh century Byzantine historian Theophylact Somicatta, as Theophilacti scolastici Simocati epistolae morales, rurales et amatoriae interpretatione Latina in 1509. He also wrote some Greek poetry himself.


Copernicus is often hailed as the first modern astronomer but as many historians have pointed out, his initial intention, following the lead of Regiomontanus, was to restore the purity of Greek astronomy, a very humanist orientated undertaking. He wanted to remove the Ptolemaic equant point, which he saw as violating the Platonic ideal of uniform circular motion. De revolutionibus was modelled on Ptolemaeus’ Mathēmatikē Syntaxis, or more accurately on the Epytoma in almagesti Ptolemei of Peuerbach and Regiomontanus.

Tycho Brahe (1546–1601) was also heavily imbued with the humanist spirit. His elaborate, purpose-built home, laboratory, and observatory on the island of Hven, Uraniborg, was built in the style of the Venetian architect Andrea Palladio (1508–1580),

Portrait of Palladio by Alessandro Maganza Source: Wikimedia Commons

the most influential of the humanist architects, and was one of the earliest buildings constructed in the Renaissance style in Norther Europe.


All of the Early Modern astronomers from Toscanelli down to at least Tycho, and very much including Copernicus, were dedicated to the humanist ideal of restoring what they saw as the glory of classical astronomy from antiquity. Only incidentally did they pave a road that led away from antiquity to modern astronomy. 


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

The Epicurean mathematician

Continuing our look at the group of mathematician astronomers associated with Nicolas-Claude Fabri de Peiresc (1580-1637) in Provence and Marin Mersenne (1588–1648) in Paris, we turn today to Pierre Gassendi (1592–1655), celebrated in the world of Early Modern philosophy, as the man who succeeded in making Epicurean atomism acceptable to the Catholic Church. 

Pierre Gassendi Source: Wikimedia Commons

Pierre Gassendi was born the son of the peasant farmer Antoine Gassend and his wife Fançoise Fabry in the Alpes-de-Haute-Provence village of Champtercier on 22 January 1592. Recognised early as something of a child prodigy in mathematics and languages, he was initially educated by his uncle Thomas Fabry, a parish priest. In 1599 he was sent to the school in Digne, a town about ten kilometres from Champtercier, where he remained until 1607, with the exception of a year spent at school in another nearby village, Riez. 

In 1607 he returned to live in Champtercier and in 1609 he entered the university of Aix-en-Provence, where his studies were concentrated on philosophy and theology, also learning Hebrew and Greek. His father Antoine died in 1611. From 1612 to 1614 his served as principle at the College in Digne. In 1615 he was awarded a doctorate in theology by the University of Avignon and was ordained a priest in 1615. From 1614 he held a minor sinecure at the Cathedral in Digne until 1635, when he was elevated to a higher sinecure. From April to November in 1615 he visited Paris for the first time on Church business. 

Cathédrale Saint-Jérome de Digne Source: Wikimedia Commons

In 1617 both the chair of philosophy and the chair of theology became vacant at the University of Aix; Gassendi applied for both chairs and was offered both, one should note that he was still only twenty-four years old. He chose the chair for philosophy leaving the chair of theology for his former teacher. He remained in Aix for the next six years. 

When Gassendi first moved to Aix he lived in the house of the Provencal astronomer Joseph Gaultier de la Valette (1564–1647), vicar general of Aix and Peiresc’s observing partner. Whilst living in Gaultier’s house he got to know Jean-Baptiste Morin (1583–1556), who was also living there as Gaultier’s astronomical assistant. Although, in later years, in Paris, Gassendi and Morin would have a major public dispute, in Aix the two still young aspiring astronomers became good friends. It was also through Gaultier that Gassendi came to the attention of Peiresc, who would go on to become his patron and mentor. 

Jean-Baptiste Morin Source: Wikimedia Commons

For the next six years Gassendi taught philosophy at the University of Aix and took part in the astronomical activities of Peiresc and Gaultier, then in 1623 the Jesuits took over the university and Gassendi and the other non-Jesuit professors were replaced by Jesuits. Gassendi entered more than twenty years of wanderings without regular employment, although he still had his sinecure at the Cathedral of Digne.

In 1623, Gassendi left Aix for Paris, where he was introduced to Marin Mersenne by Peiresc. The two would become very good friends, and as was his wont, Mersenne took on a steering function in Gassendi’s work, encouraging him to engage with and publish on various tropics. In Paris, Gassendi also became part of the circle around Pierre Dupuy (1582–1651) and his brother Jacques (1591–1656), who were keepers of the Bibliothèque du Roi, today the Bibliothèque nationale de France, and who were Ismael Boulliau’s employers for his first quarter century in Paris.

Pierre Dupuy Source: Wikimedia Commons

The Paris-Provence group Peiresc (1580–1637), Mersenne (1588–1648), Morin (1583–1656), Boulliau (1605–1694), and Gassendi (1592–1655) are all members of the transitional generation, who not only lived through the transformation of the scientific view of the cosmos from an Aristotelian-Ptolemaic geocentric one to a non-Aristotelian-Keplerian heliocentric one but were actively engaged in the discussions surrounding that transformation. When they were born in the late sixteenth century, or in Boulliau’s case the early seventeenth century, despite the fact that Copernicus’ De revolutionibus had been published several decades earlier and although a very small number had begun to accept a heliocentric model and another small number the Tychonic geo-heliocentric one, the geocentric model still ruled supreme. Kepler’s laws of planetary motion and the telescopic discoveries most associated with Galileo still lay in the future. By 1660, not long after their deaths, with once again the exception of Boulliau, who lived to witness it, the Keplerian heliocentric model had been largely accepted by the scientific community, despite there still being no empirical proof of the Earth’s movement. 

Given the Church’s official support of the Aristotelian-Ptolemaic geocentric model and after about 1620 the Tychonic geo-heliocentric model, combined with its reluctance to accept this transformation without solid empirical proof, the fact that all five of them were devout Catholics made their participation in the ongoing discussion something of a highwire act. Gassendi’s personal philosophical and scientific developments over his lifetime are a perfect illustration of this. 

During his six years as professor of philosophy at the University of Aix, Gassendi taught an Aristotelian philosophy conform with Church doctrine. However, he was already developing doubts and in 1624 he published the first of seven planned volumes criticising Aristotelian philosophy, his Exercitationes paradoxicae adversus aristoteleos, in quibus praecipua totius peripateticae doctrinae fundamenta excutiuntur, opiniones vero aut novae, aut ex vetustioribus obsoletae stabiliuntur, auctore Petro Gassendo. Grenoble: Pierre Verdier. In 1658, Laurent Anisson and Jean Baptiste Devenet published part of the second volume posthumously in Den Hague in 1658. Gassendi seems to have abandoned his plans for the other five volumes. 

To replace Aristotle, Gassendi began his promotion of the life and work of Greek atomist Epicurus (341–270 BCE). Atomism in general and Epicureanism in particular were frowned upon by the Christian Churches in general. The Epicurean belief that pleasure was the chief good in life led to its condemnation as encouraging debauchery in all its variations. Atomists, like Aristotle, believed in an eternal cosmos contradicting the Church’s teaching on the Creation. Atomist matter theory destroyed the Church’s philosophical explanation of transubstantiation, which was based on Aristotelian matter theory. Last but no means least Epicurus was viewed as being an atheist. 

In his biography of Epicurus De vita et moribus Epicuri libri octo published by Guillaume Barbier in Lyon in 1647

and revival and reinterpretation of Epicurus and Epicureanism in his Animadversiones in decimum librum Diogenis Laertii: qui est De vita, moribus, placitisque Epicuri. Continent autem Placita, quas ille treis statuit Philosophiae parteis 3 I. Canonicam, …; – II. Physicam, …; – III. Ethicam, … and his Syntagma philosophiae Epicuri cum refutationibus dogmatum quae contra fidem christianam ab eo asserta sunt published together by Guillaume Barbier in Lyon in 1649,

Gassendi presented a version of Epicurus and his work that was acceptable to Christians, leading to both a recognition of the importance of Epicurean philosophy and of atomism in the late seventeenth and early eighteenth centuries. 

Gassendi did not confine himself to work on ancient Greek philosophers. In 1629,  pushed by Mersenne, the scientific agent provocateur, he wrote an attack on the hermetic philosophy of Robert Fludd (1574–1637), who famously argued against mathematics-based science in his debate with Kepler. Also goaded by Mersenne, he read Descartes’ Meditationes de prima philosophia (Meditations on First Philosophy) (1641) and published a refutation of Descartes’ methodology. As a strong scientific empiricist, Gassendi had no time for Descartes’ rationalism. Interestingly, it was Gassendi in his Objections (1641), who first outlined the mind-body problem, reacting to Descartes’ mind-body dualism. Descartes was very dismissive of Gassendi’s criticisms in his Responses, to which Gassendi responded in his Instantiae (1642). 

Earlier, Gassendi had been a thorn in Descartes side in another philosophical debate. In 1628, Gassendi took part in his only journey outside of France, travelling through Flanders and Holland for several months, although he did travel widely throughout France during his lifetime. Whilst in Holland, he visited Isaac Beeckman (1588–1637) with whom he continued to correspond until the latter’s death. Earlier, Beeckman had had a massive influence on the young Descartes, introducing him to the mechanical philosophy. In 1630, Descartes wrote an abusive letter denying any influence on his work by Beeckman. Gassendi, also a supporter of the mechanical philosophy based on atomism, defended Beeckman.

Like the others in the Mersenne-Peiresc group, Gassendi was a student and supporter of the works of both Johannes Kepler (1571–1630) and Galileo Galilei (1564–1642) and it is here that he made most of his contributions to the evolution of the sciences in the seventeenth century. 

Having been introduced to astronomy very early in his development by Peiresc and Gaultier de la Valette, Gassendi remained an active observational astronomer all of his life. Like many others, he was a fan of Kepler’s Tabulae Rudolphinae (Rudolphine Tables) (1627) the most accurate planetary tables ever produced up till that time. Producing planetary tables and ephemerides for use in astrology, cartography, navigation, etc was regarded as the principal function of astronomy, and the superior quality of Kepler’s Tabulae Rudolphinae was a major driving force behind the acceptance of a heliocentric model of the cosmos. Consulting the Tabulae Rudolphinae Gassendi determined that there would be a transit of Mercury on 7 November 1631. Four European astronomers observed the transit, a clear proof that Mercury orbited the Sun and not the Earth, and Gassendi, who is credited with being the first to observe a transit of Mercury, published his observations Mercvrivs in sole visvs, et Venvs invisa Parisiis, anno 1631: pro voto, & admonitione Keppleri in Paris in 1632.

He also tried to observe the transit of Venus, predicted by Kepler for 6 December 1631, not realising that it was not visible from Europe, taking place there during the night. This was not yet a proof of heliocentricity, as it was explainable in both the Capellan model in which Mercury and Venus both orbit the Sun, which in turn orbits the Earth and the Tychonic model in which the five planets all orbit the Sun, which together with the Moon orbits the Earth. But it was a very positive step in the right direction. 

In his De motu impresso a motore translato. Epistolæ duæ. In quibus aliquot præcipuæ tum de motu vniuersè, tum speciatim de motu terræattributo difficulatates explicantur published in Paris in 1642, he dealt with objections to Galileo’s laws of fall.

Principally, he had someone drop stones from the mast of a moving ship to demonstrate that they conserve horizontal momentum, thus defusing the argument of those, who claimed that stones falling vertically to the Earth proved that it was not moving. In 1646 he published a second text on Galileo’s theory, De proportione qua gravia decidentia accelerantur, which corrected errors he had made in his earlier publication.

Like Mersenne before him, Gassendi tried, using a cannon, to determine the speed of sound in 1635, recording a speed of 1,473 Parian feet per second. The actual speed at 20° C is 1,055 Parian feet per second, making Gassendi’s determination almost forty percent too high. 

In 1648, Pascal, motivated by Mersenne, sent his brother-in-law up the Puy de Dôme with a primitive barometer to measure the decreasing atmospheric pressure. Gassendi provided a correct interpretation of this experiment, including the presence of a vacuum at the top of the tube. This was another indirect attack on Descartes, who maintained the assumption of the impossibility of a vacuum. 

Following his expulsion from the University of Aix, Nicolas-Claude Fabri de Peiresc’s house became Gassendi’s home base for his wanderings throughout France, with Peiresc helping to finance his scientific research and his publications. The two of them became close friends and when Peiresc died in 1637, Gassendi was distraught. He preceded to mourn his friend by writing his biography, Viri illvstris Nicolai Clavdii Fabricii de Peiresc, senatoris aqvisextiensis vita, which was published by Sebastian Cramoisy in Paris in 1641. It is considered to be the first ever complete biography of a scholar. It went through several edition and was translated into English.

In 1645, Gassendi was appointed professor of mathematics at the Collège Royal in Paris, where he lectured on astronomy and mathematics, ably assisted by the young Jean Picard (1620–1682), who later became famous for accurately determining the size of the Earth by measuring a meridian arc north of Paris.

Jean Picard

Gassendi only held the post for three years, forced to retire because of ill health in 1648. Around this time, he and Descartes became reconciled through the offices of the diplomat and cardinal César d’Estrées (1628–1714). 

Gassendi travelled to the south for his health and lived for two years in Toulon, returning to Paris in 1653 when his health improved. However, his health declined again, and he died of a lung complaint in 1655.

Although, like the others in the group, Gassendi was sympathetic to a heliocentric world view, during his time as professor he taught the now conventional geo-heliocentric astronomy approved by the Catholic Church, but also discussed the heliocentric systems. His lectures were written up and published as Institutio astronomica juxta hypotheseis tam veterum, quam Copernici et Tychonis in 1647. Although he toed the party line his treatment of the heliocentric was so sympathetic that he was reported to the Inquisition, who investigated him but raised no charges against him. Gassendi’s Institutio astronomica was very popular and proved to be a very good source for people to learn about the heliocentric system. 

As part of his campaign to promote the heliocentric world view, Gassendi also wrote biographies of Georg Peuerbach, Regiomontanus, Copernicus, and Tycho Brahe. It was the only biography of Tycho based on information from someone, who actually knew him. The text, Tychonis Brahei, eqvitis Dani, astronomorvm coryphaei vita, itemqve Nicolai Copernici, Georgii Peverbachii & Ioannis Regiomontani, celebrium Astronomorum was published in Paris in 1654, with a second edition appearing in Den Hague in the year of Gassendi’s death, 1655. In terms of historical accuracy, the biographies are to be treated with caution.

Gassendi also became engaged in a fierce dispute about astronomical models with his one-time friend from his student days, Jean-Baptiste Morin, who remained a strict geocentrist. I shall deal with this when I write a biographical sketch of Morin, who became the black sheep of the Paris-Provencal group.

Like the other members of the Paris-Provencal group, Gassendi communicated extensively with other astronomers and mathematician not only in France but throughout Europe, so his work was well known and influential both during his lifetime and also after his death. As with all the members of that group Gassendi’s life and work is a good example of the fact that science is a collective endeavour and often progresses through cooperation rather than rivalry. 

Leave a comment

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

Christmas Trilogy 2021 Part 3: Don’t throw the baby out with the bathwater 

Christmas Trilogy 2021 Part 3: Don’t throw the baby out with the bathwater 

From the beginning of European astronomy, sometime during the third millennium BCE in the Fertile Crescent, all the way down to the middle of the seventeenth century CE, nearly all active astronomers were practicing astrologers. In the Early Modern Period almost without exception the astronomers, who contributed to the birth of the new heliocentric astronomy were also astrologers, who believed in celestial influence. Even Galileo, who is falsely hailed as the founder of modern science, was a practicing astrologer and all the evidence points to the fact that he believed in it. So, what about Johannes Kepler? Kepler is a fascinating case, as for large parts of his life, he actively practiced an astrology that he didn’t believe in but believed in an astrology that nobody practiced.  

Portrait of Kepler by an unknown artist in 1620. Source: Wikimedia Commons

Kepler obviously grew up in a culture, where astrology was part of everyday life and he seemed to accept it as did almost everyone. A biographical sketch of himself and his family in which he famously describes himself as a lap dog, Ich habe in jeder Hinsicht die Natur eines Hundes. Ich bin wie ein verwöhntes Haushündchen (I have in every respect the nature of a dog. I am like a pampered lapdog). Is full of scattered astrological references. For example:

I am, however, not arrogant and contemptuous toward public opinion, but tend, admittedly, to rough speech.

When, for example, Saturn looks towards Mercury, it gets cold, so the mind droops, when Jupiter looks towards Mercury, it causes everything to become damp and fervid. There everything pushes to acquisitiveness, here to jealousy…

My father Heinrich was born on 19 January 1547, Saturn ruined everything, produced a heinous, rough, argumentative, and in the end, man of evil death.

These astrological references are, however, more than somewhat unusual, as they only refer to the influence of planets, which I will explain later. Kepler’s life was not only imbued with astrology on a daily level, astrology, also played a significant role in the various stages of his career.

When he left Tübingen University in 1594, it was to take up the position of mathematics teacher at the Protestant school in Graz. However, this job also included the position of district mathematicus, one of whose obligations was to produce the annual astrological writing calendar and prognostication for the district. The writing calendar provided the yearly astronomical and astrological data to enable physicians and barber-surgeons to practice their iatromathematics or astro-medicine, to make diagnoses and to know when the good and bad days for applying bloodletting, cupping, and purging were. Although onerous, this undertaking was a good additional source of income. The calendar writer was a paid official, but his calendars and prognostications were sold commercial, with the normal practice of the printer-publisher and the calendar writer sharing the not insubstantial profits. 

Graz by Paulus Fürst 1667

Because he successful forecast a severe winter, Turkish military advances, and peasant uprisings in his very first prognostications in Graz in 1595, he was a big hit establishing his reputation as an excellent astrologer. This would later prove helpful in a difficult situation. In 1598, the Catholic authorities in Graz, as part of the Counter Reformation, forced all the Protestants in the district to either convert to Catholicism or leave. Kepler was granted a special dispensation, not because he was a schoolteacher, the Protestant school was closed down, but because of his respected status as district astrologer. A year later even this status could not protect him, and he was forced to leave Graz for Prague. 

Initially in Prague, Kepler was Tycho Brahe’s colleague and quasi-assistant, but within a year he had replaced him as Imperial Mathematicus. In this post he was free to carry on his astronomical research, but his principal function was that of court astrologer. He was required to provide and interpret horoscopes for the Emperor Rudolf II, who believed strongly in all forms of esotericism.

Rudolf II, Copper engraving by  Egidius Sadeler, 1609

He also continued his function as producer of writing calendars and prognostications. When Rudolf lost his throne to his brother Matthias, Kepler was able to keep his title of Imperial Mathematicus, but was required to leave Prague. He now landed in Linz as district mathematicus, with, once again, district astrologer as one of his main functions. 

Linz 1497

In 1626, the Thirty Years War forced Kepler and his family to leave Linz and to seek refuge and new employment elsewhere. In 1628, he found new employment as court astrologer to the commander of the Catholic forces Albrecht von Wallenstein (1583–1634).

Albrecht von Wallenstein engraving by unknown artist c. 12625 Source: Wikimedia Commons

This was not the first time that Wallenstein, who was obsessed with astrology, had employed Kepler’s services as an astrologer. In 1608, a physician by the name of Stromair approached Kepler in Prague with the request to cast a horoscope for an anonymous noble lord. Because Stromair was a reputable physician, and probably because the payment offered was generous, Kepler always had money problems, he accepted the commission.

Wallenstein’s horoscope by Kepler 1608

Although apparently anonymous, it appears that Kepler knew, who the subject of the horoscope was and in his detailed analysis drew a very accurate portrait of the Catholic grandee. Wallenstein was very impressed and because the prognostications in the horoscope only ran until 1625, in that year he commissioned Kepler to write an extension.

Kepler’s interpretation of Wallenstein’s horoscope from 1608, as returned forvextention in 1625 with marginalia by Wallenstein Source: Wikimedia Commons

The new prognostication contained a non-specific warning for the beginning of the year 1634. Wallenstein was murdered on 25 February 1634. Kepler did not live to see the fulfilment of his prognostication having died in 1630. 

The assassination of Wallenstein in Egger/Cheb in 1634 artist unknown Source: Wikimedia Commons

Given that astrology basically financed Kepler’s existence for nearly all of his adult life, it might come as something as a surprise that he didn’t actually believe in conventional horoscope astrology. However, before anybody jumps to the conclusion that he did it just for the money, he did believe very strongly in celestial influence the basic premise on which astrology was based, an apparent contradiction.

Kepler rejected nearly the whole apparatus on which traditional Western astrology was based. He thought that the division of the ecliptic, the apparent path of the Sun around the Earth, into the twelve sun signs was purely arbitrary and had no basis in reality. He also rejected all the various schemes for dividing the zodiac into houses, for the same reason. One might ask, if Kepler rejected the whole apparatus, how did he cast so many horoscopes in his life? Being a fully trained astronomer, he could, of course, talk the talk, but he, as he tells us, filled out his interpretations with a mixture of common sense, shrewd observation, and applied psychology. Was he cheating his clients? He wasn’t doing astrology according to the book, but people were satisfied with his horoscopes. 

Although he rejected the conventional astrology, as already stated, Kepler very much believed in celestial influence and it in fact was an integral part of his entire scientific philosophy. Kepler believed that the planets radiated influence and only the planets, not the stars, not the zodiac signs, or the houses. Moreover, he believed that a single planet could not exercise influence but only two or more planets in combination, when they stood on the ecliptic at specific geometrical angles to each other, 90°, 180°, 60° etc., the so-called aspects. 

Kepler wrote several publications explaining his new astrological model, the earliest De Fundamentis Astrologiae Certioribus (Concerning the More Certain Fundamentals of Astrology) published in 1601, as a forward to his annual prognostications.

Source: Wikidata

Kepler astrology was not determinist, it only indicated tendencies and not certainties. His biggest presentation of his views on astrology came about as a result of a dispute over astrology with the German physician and astrologer, Helisäus Röslin (1545–1616), physician-in-ordinary to the count palatine of Veldenz and the count of Hanau-Lichtenberg in Buchsweiler in Alsace. The two of them had know each other since their university days. 

Röslin wrote an astrological interpretation of the nova observed in Europe in 1604. Kepler took Röslin to task, in his own publication on the nova De Stella Nova in Pede Serpentarii (On the New Star in the Foot of the Serpent Handler) in 1606.

De Stella Nova in Pede Serpentarii Source: Wikimedia Commons

Röslin responded with his Discurs von heutigen Zeit Beschaffenheit, which Kepler countered with Antwort auff Röslini Discurs, in which he also defended his heliocentric world view against Röslin’s Tychonic system. At the same time as Röslin published his Discurs von heutigen Zeit Beschaffenheit,dedicating it to Margrave Georg Friedrich von Baden, this noble’s physician-in-ordinary, Philip Feselius also dedicated to him his Discurs von de Astrologia iudiciaria. This was a total attack on astrology, which Feselius rejected completely. 

Kepler now re-entered the debated with a book, also dedicated to Margrave Georg Friedrich, Tertius Interveniens, das ist Warnung an etliche Theologos, Medicos, vnd Philosophos, sonderlich D. Philippum Feselium, dass sie bey billicher Verwerffung der Sternguckerischen Aberglauben nicht das Kindt mit dem Badt ausschütten vnd hermit  iher Profession vnwissendt zuwider handlen (1609). (Tertius Interveniens, that is warning to some theologians, medics and philosophers, especially Dr Philip Feselius, that they in cheap condemnation of the stargazer’s superstition do not throw out the baby with the bath and hereby unknowingly act contrary to their profession). Tertius Interveniens means Third-party interventions.

Tertius Intervenies

The Tertius Interveniens is a quite extraordinary publication. In 140 numbered entries that vary between a short paragraph and several pages Kepler presents a complete picture of how he sees his astronomy, astrology, natural philosophy, geometry, harmony, and theology as an integrated system. Kepler comes out swinging:

No one should consider unbelievable that there could come out of astrology foolishness and godlessness also useful cleverness and holiness out of unclean slimy substance also a snail, mussel, oyster or eel useful for eating out of the great heap of caterpillar dirt also a silk spinner and finally that out of evil-smelling dung also perhaps a good little grain yes a pearly or golden corn could be scraped for and found by an industrious hen

In some senses the Tertius Interveniens in one of the most complete presentations of Kepler’s world of thought, but because it’s in German and not Latin and because it is purely polemical and not scientific it generally gets ignored, when people discuss Kepler’s contributions to the history of science. 

In a quite extraordinary paper the historian of Early Modern science, Edward Rosen, once argued that Kepler was foolish enough as a young man to practice astrology but as he matured, he abandoned this foolishness. This is fascinating, as in the fourth book of his Harmonices mundi libri V, published in 1619, Kepler repeats the basic contents of his Tertius Interveniens, but this time more technically, that is within the context of his geometrical harmony theory. Kepler remained a convinced astrologer his whole life, but he was a reforming astrologer, who rejected nearly all of the astrology practiced by others in his times.

The first time I came across the Tertius Interveniens, I was fascinated by his usage of the expression “do not throw out the baby with the bath(water),” as this is a common phrase that is still in use, and I hadn’t thought of it being that old. I recently discovered that when Kepler used it, it was roughly a hundred years old and German in origin. The earliest known instance of the proverb, in print, “to throw the baby out with the bathwater,” is in the Narrenbeschwörung (Appeal to Fools) by Thomas Murner (1475–1537), a humanist satirist, published in Straßburg in 1512.


Filed under History of Astrology, History of Astronomy

OHMS or everything you wanted to know about the history of trigonometry and didn’t know who to ask

When I was a kid, letters from government departments came in buff, manila envelopes with OHMS printed on the front is large, black, capital letters. This acronym stood for, On Her Majesty’s Service and earlier during Liz’s father’s reign (and no I’m not that old, although I was just born in his reign), On His Majesty’s Service, implying that civil servants worked directly for the monarch.  This was, of course, the origin of the title of Ian Fleming’s eleventh James Bond novel, On Her Majesty’s Secret Service

When I started learning trigonometry at school this acronym took on a whole new meaning as a mnemonic for the sine relation in right angle triangles, Opposite over Hypotenuse Means Sine. Recently it occurred to me that we had no mnemonic for the other trigonometric relations. Now in those days or even later when the trigonometry I was taught got more complex, I wasn’t aware of the fact that this mathematical discipline had a history. Now, a long year historian of mathematics, I am very much aware of the fact that trigonometry has a very complex, more than two-thousand-year history, winding its way from ancient Greece over India, the Islamic Empire and Early Modern Europe down to the present day. 

The Canadian historian of mathematics, Glen van Brummelen has dedicated a large part of his life to researching, writing up and publishing that history of trigonometry. The results of his labours have appeared in three volumes, over the years, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry, Princeton University Press, Princeton and Oxford, 2009, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, Princeton and Oxford, 2013 and most recently The Doctrine of TrianglesA History of Modern Trigonometry, Princeton University Press, Princeton and Oxford, 2021. He describes himself as the “best trigonometry historian, and the worst trigonometry historian”, as he is the only one[1]

A review of these three volumes could be written in one sentence, if you are interested in the history of trigonometry, then these three masterful volumes are essential. One really doesn’t need to say more, but in what follows I will give a brief sketch of each of the books. 

The Mathematics of the Heavens and the Earth: The Early History of Trigonometry delivers exactly what it says on the cover. The book opens with a brief but detailed introduction to the basics of spherical astronomy, because for a large part of the period covered, what we have is not the history of plane trigonometry, that’s the stuff we all learnt at school, but spherical trigonometry, that is the geometry of triangles on the surface of a sphere, which was developed precisely to do spherical astronomy. 

A friendly warning for potential readers this is not popular history but real, hardcore history of mathematics with lots of real mathematical examples worked through in detail. However, given the way Van Brummelen structures his narrative, it is possible to skip the worked examples and still get a strong impression of the historical evolution of the discipline. This is possible because Van Brummelen gives a threefold description of every topic that he elucidates. First comes a narrative, fairly non-technical, description of the topic he is discussing. This is followed by an English translation of a worked example from the historical text under discussion, followed in turn by a technical explication of the text in question in modern terminology. Van Brummelen’s narrative style is clear and straightforward meaning that the non-expert reader can get good understanding of the points being made, without necessarily wading through the intricacies of the piece of mathematics under discussion. 

The book precedes chronologically. The first chapter, Precursors, starts by defining what trigonometry is and also what it isn’t. Having dealt with the definitions, Van Brummelen moves onto the history proper dealing with things that preceded the invention of trigonometry, which are closely related but are not trigonometry. 

Moving on to Alexandrian Greece, Van Brummelen takes the reader through the beginnings of trigonometry starting with Hipparchus, who produced the first chord table linking angles to chords and arcs of circles, Moving on through Theodosius of Bithynia and Menelaus of Alexandria and the emergence of spherical trigonometry. He then arrives at Ptolemy his astronomy and geography. Ptolemy gets the longest section of the book, which given that everything that follows in some way flows from his work in logical. Here we also get two defining features of the book. The problem of calculating trigonometrical tables and what each astronomer or mathematician contributed to this problem and the trigonometrical formulas that each of them developed to facilitate calculations. 

From Greece we move to India and the halving of Hipparchus’ and Ptolemy’s chords to produce the sine function and later the cosine that we still use today. Van Brummelen takes his reader step for step and mathematician for mathematician through the developments of trigonometry in India. 

The Islamic astronomers took over the baton from the Indians and continued the developments both in astronomy and geography. It was Islamic mathematicians, who developed the plane trigonometry that we know today rather than the spherical trigonometry. As with much other mathematics and science, trigonometry came into medieval Europe through the translation movement out of Arabic into Latin. Van Brummelen traces the development in medieval Europe down to the first Viennese School of mathematics, John of Gmunden, Peuerbach, and Regiomontanus. This volume closes with Johannes Werner and Copernicus, with a promise of a second volume. 

In the book itself, the brief sketch above is filled out in incredible detail covering all aspects of the evolution of the discipline, the problems, the advances, the stumbling stones and the mathematicians and astronomers, who discovered each problem, solved, or failed to solve them. To call Van Brummelen comprehensive would almost be an understatement. Having finished this first volume, I eagerly awaited the promised second volume, but something else came along instead.

Having made clear in his first book that the emphasis is very much on spherical trigonometry rather than plane trigonometry, in his second book Van Brummelen sets out to explain to the modern reader what exactly spherical trigonometry is, as it ceased to be part of the curriculum sometime in the modern period. What we have in Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry is a spherical trigonometry textbook written from a historical perspective. The whole volume is written in a much lighter and more accessible tone than The Mathematics of the Heavens and the Earth. After a preface elucidating the purpose of the book there follow two chapters, Heavenly Mathematics and Exploring the Sphere, which lay out and explain the basics in clear and easy to follow steps.

Next up, we have the historical part of the book with one chapter each on The Ancient Approach and The Medieval Approach. These chapters could be used as an aid to help understand the relevant sections of the authors first book. But fear not the reader must not don his medieval personality to find their way around the complexities of spherical trigonometry because following this historical guide we are led into the modern textbook.

The bulk of the book consists of five chapters, each of which deals in a modern style with an aspect of spherical trigonometry: Right Angle Triangles, Oblique Triangles, Areas, Angles and Polyhedra, Stereographic Projection, and finally Navigation by the Stars. The chapter on stereographic projection is particularly interesting for those involved with astrolabes and/or cartography. 

The book closes with three useful appendices. The first is on Ptolemy’s determination of the position of sun. The second is a bibliography of textbooks on or including spherical trigonometry with the very helpful indication, which of them are available on Google Books. The final appendix is a chapter by chapter annotated list of further reading on each topic. 

If you wish to up your Renaissance astrology game and use the method of directions to determine your date of death, which require spherical trigonometry to convert from one celestial coordinate system to another, then this is definitely the book for you. It is of course also a brilliant introduction for anybody, who wishes to learn the ins and outs of spherical trigonometry. 

I bought Van Brummelen’s first book when it was published, in 2009, and read it with great enthusiasm, but experienced a sort of coitus interruptus, when in stopped in the middle of the Renaissance, the period that interested me most. I was consoled by the author’s declaration that a second volume would follow, which I looked forward to with great expectations. Over the years those expectations dimmed, and I began to fear that the promised second volume would never appear, so I was overjoyed when the publication of The Doctrine of Triangles was announced this year and immediately placed an advanced order. I was not disappointed. 

The modern history of trigonometry continues where the early history left off, tracing the developments of trigonometry in Europe from Regiomontanus down to Clavius and Gunter in the early seventeenth century. There then follows a major change of tack, as Van Brummelen delves into the origins of logarithms.

Today in the age of the computer and the pocket calculator, logarithmic tables are virtually unknown, a forgotten relic of times past. I, however, grew up using my trusty four figure log tables to facilitate calculations in maths, physics, and chemistry. Now, school kids only know logarithms as functions in analysis. One thing that many, who had the pleasure of using log tables, don’t know is that Napier’s first tables were of the logarithms of trigonometrical factions in order to turn the difficult multiplications and divisions of sines, cosines et al in spherical trigonometry into much simpler additions and subtractions and therefore Van Brummelen’s detailed presentation of the topic.

Moving on, in his third chapter, Van Brummelen now turns to the transition of trigonometry as a calculation aid in spherical and plane triangles to trigonometrical functions in calculus. There where they exist in school mathematics today. Starting in the period before Leibniz and Newton, he takes us all the way through to Leonard Euler in the middle of the eighteenth century. 

The book now undergoes a truly major change of tack, as Van Brummelen introduces a comparative study of the history of trigonometry in Chinese mathematics. In this section he deals with the Indian and Islamic introduction of trigonometry into China and its impact. How the Chinese dealt with triangles before they came into contact with trigonometry and then the Jesuit introductions of both trigonometry and logarithms into China and to what extent this influenced Chinese geometry of the triangle. A fascinating study and an enrichment of his already excellent book.

The final section of the book deals with a potpourri of developments in trigonometry in Europe post Euler. To quote Van Brummelen, “A collection of short stories is thus more appropriate here than a continuous narrative.” The second volume of Van Brummelen’s history is just as detailed and comprehensive as the first. 

All three of the books display the same high level of academic rigour and excellence. The two history volumes have copious footnotes, very extensive bibliographies, and equally extensive indexes. The books are all richly illustrated with many first-class explanatory diagrams and greyscale prints of historical title pages and other elements of the books that Van Brummelen describes. All in all, in his three volumes Van Brummelen delivers a pinnacle in the history of mathematics that sets standards for all other historians of the discipline. He really does live up to his claim to be “the best historian of trigonometry” and not just because he’s the only one.

Coda: If the potential reader feels intimidated by the prospect of the eight hundred and sixty plus pages of the three volumes described here, they could find a gentle entry to the topic in Trigonometry: A Very Short Introduction (OUP, 2020), which is also authored by Van Brummelen, a sort of Van Brummelen light or Van Brummelen’s greatest hits.

In this he covers a wide range of trigonometrical topics putting them into their historical context. But beware, reading the Very Short Introduction could well lead to further consumption of Van Brummelen’s excellent work. 

[1] This is not strictly true as Van Brummelen has at least two predecessors both of who he quotes in his works. The German historian Anton von Braunmühl, who wrote several articles and a two volume Vorlesung über Geschichte der Trigonometrie (Leipzig, 1900/1903) and the American Sister Mary Claudia Zeller, The Development of Trigonometry from Regiomontanus to Pitiscus (Ann Arbor 1944)


Filed under History of Astronomy, History of Cartography, History of Islamic Science, History of Mathematics, History of Navigation

The astronomical librarian 

I’m continuing my look at the French mathematician astronomers of the seventeenth century with some of those, who were both members of Nicolas-Claude Fabri de Peiresc’s group of telescopic, astronomical observers, as well as Marin Mersenne’s informal Academia Parisiensis, starting with Ismael Boulliau (1605–1694), who like Peiresc and Mersenne was also a prominent member of the Republic of Letters with about 5000 surviving letters. 

Ismael Boulliau Source: Wikimedia Commons

Boulliau was born in Loudun, France the second son of Ismael Boulliau a notary and amateur astronomer and Susanne Motet on 28 September 1605. The first son had been born a year earlier and was also named Ismael, but he died and so the name was transferred to their second son. Both of his parents were Calvinists. His father introduced him to astronomy and in his Astronomia philolaica (1645) Ismael junior tells us that his father observed both Halley’s comet in 1607 and the great comet of 1618. The later was when Boulliau was thirteen years old, and one can assume that he observed together with his father. 

Probably following in his father’s footsteps, he studied law but at the age of twenty-one he converted to Catholicism and in 1631, aged twenty-six, he was ordained a priest. In 1632 he moved to Paris and began to work for Pierre Dupuy (1582–1651) and his brother Jacques (1591–1656), who were keepers of the Bibliothèque du Roi, today the Bibliothèque nationale de France. Boulliau held this position until the death of the Dupuy brothers and during that time travelled widely in Europe collecting books and manuscripts for the library. 

Pierre Dupuy Source: Wikimedia Commons

Boulliau also enjoyed the patronage of the powerful and influential de Trou family, who were closely connected with the library and who financed his book collecting travels. Following the death of the Dupuy brothers he became employed by the French ambassador to the United Provinces, a member of the de Trou family, a secretary and librarian. In 1666, following a dispute with his employer, he became librarian at the Collège de Laon in Paris. For the last five years of his live he returned to the priesthood in the Abbey St Victor near Paris where he died aged 89. Although Boulliau was an active member of Mersenne’s Academia Parisiensis he never became a member of the Académie des sciences, but he was elected one of the first foreign associates of the Royal Society on 4 April 1667. 

Abbey of St. Victor, 1655 Source: Wikimedia Commons

 Like Peiresc, Boulliau was a polymath with extensive knowledge of a wide range of humanities topics, which was useful in his work as a librarian, but, as with Peiresc, it is scientific activities that are of interest here. He continued to make astronomical observations throughout his life, which were of a high level of accuracy. In his Principia, Newton puts him on a level with Kepler for his determination of the planetary orbits. In Book 3 Phenomenon 4 of Principia Newton writes: 

But of all astronomers, Kepler and Boulliau have determined the magnitude of the orbits from observations with the most diligence. 

Boulliau’s first significant scientific publication was, however, not in astronomy but in optics, his De natura lucis (On the Nature of Light) (1638) based on the discussions he was having with Gassendi on the topic. This work is not particular important in the history of optics but it does contain his discussion of Kepler’s inverse square law for the propagation of light.

Source: Wikimedia Commons

His first astronomical work Philolaus (1639), which places him firmly in the Copernican heliocentric camp but not, yet a Keplerian was next. 

He now changed tack once again with a historical mathematical work. In 1644, he translated and published the first printed edition of Theon of Smyrna’s Expositio rerum mathematicarum ad legendum Platonem utilium a general handbook for students of mathematics of no real significance. Continuing with his mathematical publications. In 1657, he published De lineis spiralibus (On Spirals) related to the work of Archimedes and Pappus on the topic.

Source: Wikimedia Commons

Much later in 1682, he published Opus novum ad arithmeticam infinitorum, which he claimed clarified the Arithmetica infinitorum(1656) of John Wallis (1616–1703).

Source: Wikimedia Commons

All of Boulliau’s work was old fashioned and geometrical. He rejected the new developments in analytical mathematics and never acknowledged Descartes’ analytical geometry. As we shall see, his astronomy was also strictly geometrical. He even criticised Kepler for being a bad geometer. 

Boulliau’s most important publication was his second astronomical text Astronomia philolaica (1645).

Source: Wikimedia Commons

In this highly influential work, he fully accepted Kepler’s elliptical orbits but rejects almost all of the rest of Kepler’s theories. As stated above his planetary hypothesis is strictly geometrical and centres round his conical hypothesis:

“The Planets, according to that astronomer [Boulliau], always revolve in circles; for that being the most perfect figure, it is impossible they should revolve in any other. No one of them, however, continues to move in any one circle, but is perpetually passing from one to another, through an infinite number of circles, in the course of each revolution; for an ellipse, said he, is an oblique section of a cone, and in a cone, betwixt the vertices of the ellipse there is an infinite number of circles, out of the infinitely small portions of which the elliptical line is compounded. The Planet, therefore, which moves in this line, is, in every point of it, moving in an infinitely small portion of a certain circle. The motion of each Planet, too, according to him, was necessarily, for the same reason, perfectly equable. An equable motion being the most perfect of all motions. It was not, however, in the elliptical line, that it was equable, but in any one of the circles that were parallel to the base of that cone, by whose section this elliptical line had been formed: for, if a ray was extended from the Planet to any one of those circles, and carried along by its periodical motion, it would cut off equal portions of that circle in equal times; another most fantastical equalizing circle, supported by no other foundation besides the frivolous connection betwixt a cone and an ellipse, and recommended by nothing but the natural passion for circular orbits and equable motions,” (Adam Smith, History of Astronomy, IV.55-57).

Boulliau’s Conical Hypothesis [RA Hatch] Source: Wikimedia Commons

Boulliau’s theory replaces Kepler’s second law, and this led to the Boulliau-Ward debate on the topic with the English astronomer Seth Ward (1617–1689), the Savilian Professor of astronomy at Oxford University.

Bishop Seth Ward, portrait by John Greenhill Source: Wikimedia Commons

Ward criticised Boulliau’s theory in his In Ismaelis Bullialdi astro-nomiae philolaicae fundamenta inquisitio brevis (1653), also pointing out mathematical errors in Boulliau’s work. 

Boulliau responded to Ward’s criticisms in 1657, acknowledging the errors and correcting but in turn criticising Ward’s model in his De lineis spiralibus. A year earlier Ward had published his own version of Keplerian astronomy in his Astronomia geometrica (1656).

Source: Wikimedia Commons

This exchange failed to find a resolution but this very public debate between two of Europe’s leading astronomers very much raised awareness of Kepler’s work and was factor in its eventual acceptance of Kepler’s elliptical heliocentric astronomy. 

It was in his Astronomia philolaica that Boulliau was the first to form an inverse squared theory of attraction between the sun and the planets. 

As for the power by which the Sun seizes or holds the planets, and which, being corporeal, functions in the manner of hands, it is emitted in straight lines throughout the whole extent of the world, and like the species of the Sun, it turns with the body of the Sun; now, seeing that it is corporeal, it becomes weaker and attenuated at a greater distance or interval, and the ratio of its decrease in strength is the same as in the case of light, namely, the duplicate proportion, but inversely, of the distances that is, 1/d2 ​.

Here we see the influence of Kepler’s theory of light propagation, which as noted Boulliau discussed in his De natura lucis. However, having set up this hypothesis Boulliau goes on to reject it. 

… I say that the Sun is moved by its own form around its axis, by which form it was ignited and made light, indeed I say that no kind of motion presses upon the remaining planets … indeed [I say] that the individual planets are driven round by individual forms with which they were provided …

Despite Boulliau’s rejection of his own hypothesis, during Newton’s dispute with Hooke over who should get credit for the theory of gravity, he gives Boulliau the credit in a letter to Edmond Halley.

…so Bullialdus [i.e., Boulliau] wrote that all force respecting ye Sun as its center & depending on matter must be reciprocally in a duplicate ratio of ye distance from ye center, & used that very argument for it by wch you, Sr, in the last Transactions have proved this ratio in gravity. Now if Mr Hook from this general Proposition in Bullialdus might learn ye proportion in gravity, why must this proportion here go for his invention?

In 1667, Boulliau published a final astronomy book, Ad astronomos monita duo in which he was the first to establish the periodicity of the variable star, Mira Ceti.


His estimate of the period 333 days was only out by a one day. Mira had first been recognised as a variable star by David Fabricius beginning 3 August 1596.

Apart from his publications Boulliau kept Mersenne’s correspondence network alive for another thirty years after Mersenne’s death, communicating with Leopoldo de’ Medici (1617–1675) in Italy, Johannes Hevelius (1611–1687) in Danzig and Christiaan Huygens (1629–1695). Huygens first imparted his discovery of the rings of Saturn to Boulliau and Boulliau distributed Huygens’ System sarturnium (1658) in Paris. Boulliau also distributed Pascal’s Letters D’Amos Dettonville (1658–1659) to English and Dutch mathematicians, his challenge on the mathematics of the cycloid, an important publication in the development of calculus.

Ismael Boulliau is a prime example of a scholar, who didn’t make any major discoveries or develop any major theories himself but still had a very significant influence on the development of science.

Leave a comment

Filed under History of Astronomy, History of Mathematics, History of Optics, History of science, The Paris Provencal Connection

Renaissance Science – XXII

Perhaps surprisingly, land surveying as we know it today, a mathematical discipline utilising complex technological measuring instruments is very much a product of the practical mathematics of the Renaissance. Why surprisingly? Surveying is an ancient discipline that has its origins in humanity becoming settled many thousands of years ago. Ancient monuments such as the pyramids or Stonehenge definitely required some level of surveying in their construction and there are surviving documents from all literate ancient societies that refer to methods or the practice of surveying. 

All surveying uses some aspects of geometry and as Herodotus famously claimed geometry (Greek: geōmetría from geōmétrēs), which literally means measurement of earth or land, had its origins in Egyptian surveying for tax purposes. According to his account, King Sesostris divided all the lands in Egypt amongst its inhabitants in return for an annual rent. However, every year the Nile floods washing away the parts of the plots:

The country is converted into a sea, and nothing appears but the cities, which looked like islands in the Aegean. 

Those whose land had been lost objected to paying the rent, so Sesostris summoned those affected to appear before him.

Upon which, the king sent persons to examine, and determine by measurement the exact extent of the loss: and thenceforth only such a rent was demanded of him as was proportionate to the reduced size of his land. From this practice, I think, geometry first came to be known in Egypt, whence it passed into Greece.

According to legend, both Thales and Pythagoras, are reputed to have learnt their geometry in Egypt.

In all early cultures surveying was fairly primitive with measurements being made with ropes and measuring rods. In Egypt, surveyors were known as rope stretchers (harpedonaptai), the ropes used for measuring being stretched to avoid sagging.

A rope being used to measure fields. Taken from the Tomb of Menna, TT69. (c. 1500–1200 BCE) Source: Wikimedia Commons

Longer distances were either measured by estimation or by pacing. In ancient Egypt and Greece Bematistae (step measurer) where trained to walk with equal length paces and the historical records of Alexander the Great’s campaigns suggest that they were indeed highly accurate. This measuring of distances by pacing in reflected in our word mile, which is the Latin word for a thousand, mille, meaning a thousand paces.

The Latin for surveyor was agrimensores, meaning field measurers. They were also called gromatici after the groma a surveyor’s pole, an early instrument for determining lines at right angles to each other. 

The groma or gruma was a Roman surveying instrument. It comprised a vertical staff with horizontal cross-pieces mounted at right angles on a bracket. Each cross piece had a plumb line hanging vertically at each end. It was used to survey straight lines and right angles, thence squares or rectangles. They were stabilized on the high ground and pointed in the direction it was going to be used. The helper would step back 100 steps and place a pole. The surveyor would tell him where to move the pole and the helper would set it down.

(Lewis, M. J. T., Surveying instruments of Greece and Rome, McGraw Hill Professional, 2001, p. 120)
Staking out a right angle using a groma

Another instrument used for the same purpose was the dioptra. The dioptra was a sighting tube or, alternatively an alidade, that is a rod with a sight at each end, attached to a stand. If fitted with protractors, it could be used to measure angles. Hero from Alexandria wrote a whole book on this instrument and its use but there are doubts that the dioptra in the complex form described by Hero was actually used in field surveying.

Dioptra as described by Hero of Alexandria Source: Wikimedia Commons

The methods used by the Romans in field surveying were described in the works of technical authors such as Sextus Julius Frontinus (c. 40–103 CE) and Gaius Julius Hyginus (c. 64 BCE–17 CE).

All of the surveying described in antiquity was fairly small scale–measuring fields, determining boundaries, laying out military camps, etc–and geometrically centred on squares and rectangles. Cartography was done using astronomical determinations of latitude and longitude, whereby the latter was difficult, and distances estimated or paced. Nothing really changed in Europe during the medieval period. The surveying that was done was carried out using the same methods that the Romans had used. However, during the fifteenth century things began to change substantially and the first question is why?

The rediscovery of Ptolemaeus’ Geographia at the beginning of the fifteenth century, as described here, and the subsequent substantial increase in cartographical activity, as described here, played a major role, but as already stated above Ptolemaic cartography relied almost exclusively on astronomical methods and did not utilise field surveying. However, there was an increased demand for internal accuracy in maps that astronomical methods could not supply. Secondly, changes in land ownership led to an increased demand for accurate field surveying of estates that required more sophisticated methods than those of the agrimensores. Lastly, we have a good example of the knowledge crossover, typical for the Renaissance, as described in Episode V of this series. The surveyors of antiquity were artisans producing practical knowledge for everyday usage. In the Renaissance, university educated scholars began to interest themselves for this practical knowledge and make contributions to its development and it is these developments that we will now look at. 

The biggest change in surveying was the introduction of the simple geometrical figure the triangle into surveying, as Sebastian Münster, one of the most influential cosmographers (today we would say geographer) of the period, wrote in a German edition of his Cosmographia. Beschreibung aller Lender durch Sebastianum Münsterum in 1550:

Every thing you measure must be measured in triangles.

Actually, the theory of similar triangles, as explained in Euclid’s Elements, had been used in surveying in antiquity, in particular to determine the height of things or for example the width of a river. A method that I learnt as a teenager in the Boy Scouts.

What was new as we will see was the way that triangles were being used in surveying and that now it was the angles of the triangles that were measured and not the length of the sides, as in the similar triangles’ usage. We are heading towards the invention and usage of triangulation in surveying and cartography, a long-drawn-out process.

In his Ludi rerum mathematicarum (c. 1445), the architect Leon Battista Alberti describes a method of surveying by taking angular bearings of prominent points in the area he is surveying using a self-made circular protractor to create a network of triangles. He concludes by explaining that one only needs to the length of one side of one triangle to determine all the others. What we have here is an early description of a plane table surveying (see below) and step towards triangulation that, however, only existed in manuscript 

Alberti Ludi rerum mathematicarum 

Münster learnt his geometry from Johannes Stöffler (1452–1531), professor for mathematics in Tübingen, who published the earliest description of practical geometry for surveyors. In his De geometricis mensurationibus rerum (1513),

Johannes Stöffler Engraving from the workshop of Theodor de Brys, Source: Wikimedia Commons

Stöffler explained how inaccessible distances could be measured by measuring one side of a triangle using a measuring rod (pertica) and then observing the angles from either end of the measured stretch. However, most of the examples in his book are still based on the Euclidian concept of similar triangles rather than triangulation. In 1522, the printer publisher Joseph Köbel, who had published the Latin original, published a German version of Stöffler’s geometry book. 

Joseph Köbel Source: Wikimedia Commons

Both Peter Apian in his Cosmographia (1524) and Oronce Fine in his De geometria practica (1530) give examples of using triangles to measure distances in the same way as Stöffler.


Fine indicating that he knew of Stöffler’s book. Apian explicitly uses trigonometry to resolve his triangles rather than Euclidian geometry. Trigonometry had already been known in Europe in the Middle Ages but hadn’t been used before the sixteenth century in surveying. Fine, however, still predominantly used Euclidian methods in his work, although he also, to some extent, used trigonometry.

A very major development was the publication in 1533 of Libellus de locorum describendum ratione (Booklet concerning a way of describing places) by Gemma Frisius as an appendix to the third edition of Apian’s Cosmographia, which he edited, as he would all edition except the first. Here we have a full technical description of triangulation published for the first time. It would be included in all further editions in Latin, Spanish, French, Flemish, in what was the most popular and biggest selling manual on mapmaking and instrument making in the sixteenth and seventeenth centuries.

Source: Wikimedia Commons

1533 also saw the publication in Nürnberg by Johannes Petreius (c. 1497–1550) of Regiomontanus’s De triangulis omnimodis (On triangles of every kind) edited by the mapmaker and globe maker, Johannes Schöner (1477–1547).


This volume was originally written in 1464 but Regiomontanus died before he could print and publish it himself, although he had every intention of doing so. This was the first comprehensive work on trigonometry in Europe in the Early Modern Period, although it doesn’t cover the tangent, which Regiomontanus handled in his Tabula directionum (written 1467, published 1490), an immensely popular and oft republished work on astrology. 

Regiomontanus built on previous medieval works on trigonometry and the publication of his book introduces what Ivor Grattan Guinness has termed The Age of Trigonometry. In the sixteenth century it was followed by Rheticus’ separate publication of the trigonometrical section of Copernicus’s De revolutionibus, as De lateribus et angulis triangulorium in 1542. Rheticus (1514–1574) followed this in 1551 with his own Canon doctrinae triangulorum. This was the first work to cover all six trigonometric functions and the first to relate the function directly to triangles rather than circular arcs.

Source: Wikimedia Commons

Rheticus spent the rest of his life working on his monumental Opus Palatinum de Triangulis, which was, however, first published posthumously by his student Lucius Valentin Otho in 1596. Rheticus and Otho were pipped at the post by Bartholomaeus Pitiscus (1561–1613), whose Trigonometriasive de solutione triangulorum tractatus brevis et perspicuous was published in 1595 and gave the discipline its name.

Source: Wikimedia Commons

Pitiscus’ work went through several edition and he also edited and published improved and corrected editions of Rheticus’ trigonometry volumes. 

Through Gemma Frisius’ detailed description of triangulation and sixteenth century works on trigonometry, Renaissance surveyors and mapmakers now had the mathematical tools for a new approach to surveying. What they now needed were the mathematical instruments to measure distances and angles in the field and they were not slow in coming.

The measure a straight line of a given distance as a base line in triangulation surveyors still relied on the tools already used in antiquity the rope and the measuring rod. Ropes were less accurate because of elasticity and sagging if used for longer stretches. In the late sixteenth century, they began to be replaced by the surveyor’s chain, made of metal links but this also suffered from the problem of sagging due to its weight, so for accuracy wooden rods were preferred. 

A Gunter chain photographed at Campus Martius Museum. Source: Wikimedia Commons

In English the surveyor’s chain is usually referred to as Gunter’s chain after the English practical mathematician Edmund Gunter (1581–1626) and he is also often referred to erroneously as the inventor of the surveyor’s chain but there are references to the use of the surveyor’s chain in 1579, when Gunter was still a child. 

He did, however, produce what became a standardised English chain of 100 links, 66 feet or four poles, perches, or rods long, as John Ogilby (1600–1676) wrote in his Britannia Atlas in 1675:

…a Word or two of Dimensurators or Measuring Instruments, whereof the mosts usual has been the Chain, and the common length for English Measures 4 Poles, as answering indifferently to the Englishs Mile and Acre, 10 such Chains in length making a Furlong, and 10 single square Chains an Acre, so that a square Mile contains 640 square Acres…’

An English mile of 5280 feet was thus 80 chains in length and there are 10 chains to a furlong. An acre was 10 square chains. I actually learnt this antiquated system of measurement whilst still at primary school. The name perch is a corruption of the Roman name for the surveyor’s rod the pertica. 

To measure angles mapmakers and surveyors initially adopted the instruments developed and used by astronomers, the Jacob staff, the quadrant, and the astrolabe. An instrument rarely still used in astronomy but popular in surveying was the triquetum of Dreistab. The surveyors triquetum consists of three arms pivoted at two points with circular protractors added at the joints to measure angles and with a magnetic compass on the side to determine bearings. 

Surveyors then began to develop variants of the dioptra. The most notable of these, that is still in use today albeit highly modernised, was the theodolite, an instrument with sights capable of measuring angles both vertically and horizontally. The name first occurs in the surveying manual A geometric practice named Pantometria by Leonard Digges (c. 1515–c. 1559) published posthumously by his son Thomas (c. 1546–1595) in 1571.

Leonard Digges  A geometric practice named Pantometria Source

However, Digges’ instrument of this name could only measure horizontal angles. He described another instrument that could measure both vertical and horizontal angles that he called a topographicall instrument. Josua Habermehl, about whom nothing is known, but who was probably a relative of famous instrument maker Erasmus Habermehl (c. 1538–1606), produced the earliest known instrument similar to the modern theodolite, including a compass and tripod, in 1576. In 1725, Jonathan Sisson (1690–1747) constructed the first theodolite with a sighting telescope.

Theodolite 1590 Source:

A simpler alternative to the theodolite for measuring horizontal angles was the circumferentor. This was a large compass mounted on a plate with sights, with which angles were measured by taking their compass bearings.

18th century circumferentor

Instruments like the triquetum and the circumferentor were most often used in conjunction of another new invention, the plane table. Gemma Frisius had already warned in his Libellus de locorum describendum rationeof the difficulties of determining the lengths of the sides of the triangles in triangulation using trigonometry and had described a system very similar to the plane table in which the necessity for these calculation is eliminated. 

Surveying with plane table and surveyor’s chain

The plane table is a drawing board mounted on a tripod, with an alidade. Using a plumb bob, the table is centred on one end of a baseline, levelled by eye or after its invention (before 1661) with a spirit level, and orientated with a compass. The alidade is placed on the corresponding end of the scaled down baseline on the paper on the table and bearings are taken of various prominent features in the area, the sight lines being drawn directly on the paper. This procedure is repeated at the other end of the baseline creating triangles locating the prominent figures on the paper without having to calculate.

Philippe Danfrie (c.1532–1606) Surveying with a plane table

As with the theodolite there is no certain knowledge who invented the plane table. Some sources attribute the invention of the plane table to Johannes Praetorius (1537–1616), professor for mathematics at the University of Altdorf, as claimed by his student Daniel Schwentner (1585–1636). However, there was already a description of the plane table in “Usage et description de l’holomètre”, by Abel Foullon (c. 1514–1563) published in Paris in 1551. It is obvious from his description that Foullon hadn’t invented the plane table himself. 

The plane table is used for small surveys rather than mapmaking on a large scale and is not triangulation as described by Gemma Frisius. Although the Renaissance provided the wherewithal for full triangulation, it didn’t actually get used much for mapping before the eighteenth century. At the end of the sixteenth century Tycho Brahe carried out a triangulation of his island of Hven, but the results were never published. The most notable early use was by Willebrord Snel (1580–1626) to measure one degree of latitude in order to determine the size of the earth in 1615. He published the result in his Eratosthenes batavus in Leiden in 1617. He then extended his triangulation to cover much of the Netherlands.

Snel’s Triangulation of the Dutch Republic from 1615 Source: Wikimedia Commons

In the late seventeenth century Jean Picard (1620–1682) made a much longer meridian measurement in France using triangulation. 

Picard’s triangulation and his instruments

In fourteen hundred European surveyors were still using the same methods of surveying as the Romans a thousand years earlier but by the end of the seventeenth century when Jean-Dominique Cassini (1625–1712) began the mapping of France that would occupy four generations of the Cassini family for most of the eighteenth century, they did so with the fully developed trigonometry-based triangulation that had been developed over the intervening three hundred years. 


Filed under History of Astronomy, History of Cartography, History of Geodesy, History of Mathematics, History of science, Renaissance Science

The amateur, astronomical, antiquarian aristocrat from Aix

In a recent blog post about the Minim friar, Marin Mersenne (1588–1648), I mentioned that when Mersenne arrived in Paris in 1619 he was introduced to the intellectual elite of the city by Nicolas-Claude Fabri de Peiresc (1580-1637). In another recent post on the Republic of Letters I also mentioned that Peiresc was probably, the periods most prolific correspondent, with more than ten thousand surviving letters. So, who was this champion letter writer and what role did he play in the European scientific community in the first third of the seventeenth century?

Nicolas-Claude Fabri de Peiresc by Louis Finson Source: Wikimedia Commons

Nicolas-Claude Fabri was born, into a family of lawyers and politicians, in the town Belgentier near Toulon on 1 December in 1580, where his parents had fled to from their hometown of Aix-en-Provence to escape the plagues. He was educated at Aix-en-Provence, Avignon, and the Jesuit College at Tournon. Having completed his schooling, he set off to Padua in Italy, nominally to study law, but he devoted the three years, 1600–1602, to a wide-ranging, encyclopaedic study of the history of the world and everything in it. 

In this he was aided in that he became a protégé of Gian Vincenzo Pinelli (1535–1601) a humanist scholar and book collector, his library numbered about 8,500 printed works, with all-embracing intellectual interests, whose main areas were botany, optics, and mathematical instruments.

Gian Vincènzo Pinelli Source: Rijksmuseum via Wikimedia Commons

Pinelli introduced Fabri to many leading scholars including Marcus Welser (1558–1614), Paolo Sarpi (1552–1623) and indirectly Joseph Scaliger (1540–1609). Pinelli also introduced him to another of his protégés, Galileo Galilei (1564–1642). One should always remember that although he was thirty-eight years old in 1602, Galileo was a virtually unknown professor of mathematics in Padua. When Pinelli died, Fabri was living in his house and became involved in sorting his papers.

In 1602, Fabri returned to Aix-en-Provence and completed his law degree, graduating in 1604. In the same year he assumed the name Peiresc, it came from a domain in the Alpes-de-Haute-Provence, which he had inherited from his father. He never actually visited Peiresc, now spelt Peyresq.

Village of Peyresq Source: Wikimedia Commons

Following graduation Peiresc travelled to the Netherlands and England via Paris, where he made the acquaintance of other notable scholars, including actually meeting Scaliger and also meeting the English antiquarian and historian William Camden (1551–1623).

Returning to Provence, in 1607, he took over his uncle’s position as conseiller to the Parliament of Provence under his patron Guillaume du Vair (1556–1621), cleric, lawyer, humanist scholar and president of the parliament.

Guillaume-du-Vair Source: Wikimedia Commons

In 1615 he returned to Paris with du Vair as his secretary, as du Vair was appointed keeper of the seals during the regency of Marie de’ Medici (1575–1642). Peiresc continued to make new contacts with leading figures from the world of scholarship, and the arts, including Peter Paul Rubens (1577–1640).

Peter Paul Rubens self-portrait 1623

Peiresc acted as a go between in the negotiations between Reubens and the French court in the commissioning of his Marie de’ Medici Cycle. Just one of Peiresc’s many acts of patronage in the fine arts.

Marie de’ Medici Cycle in the Richelieu wing of the Louvre Source: Wikimedia Commons

In 1621 de Vair died and in 1623 Peiresc returned to Provence, where he continued to serve in the parliament until his death in 1637.

Peiresc was an active scholar and patron over a wide range of intellectual activities, corresponding with a vast spectrum of Europe’s intellectual elite, but we are interested here in his activities as an astronomer. Having developed an interest for astronomical instruments during his time as Pinelli’s protégé, Peiresc’s astronomical activities were sparked by news of Galileo’s telescopic discoveries, which reached him before he got a chance to read the Sidereus Nuncius. He rectified this lack of direct knowledge by ordering a copy from Venice and borrowing one from a friend until his own copy arrived.

Source: Wikimedia Commons

He immediately began trying to construct a telescope to confirm or refute Galileo’s claims, in particular the discovery of the first four moons of Jupiter. At this point in his life Peiresc was still a geocentrist, later he became a convinced heliocentrist. We know very little about where and how he acquired his lenses, but we do know that he had various failures before he finally succeeded in observing the moons of Jupiter for himself, in November 1610. In this he was beaten to the punch by his fellow Provencal astronomer Joseph Gaultier de la Valette (1564–1647), vicar general of Aix. At this point it is not clear whether the two were competing or cooperating, as they would then later do with Gaultier de la Valette becoming a member of Peiresc’s Provencal astronomical observation group. Shortly thereafter, Peiresc became the first astronomer to make telescopic observations of the Orion Nebular and Gaultier de la Valette the second. This is rather strange as the Orion Nebular is visible to the naked eye. However, apparently none of the telescopic astronomy pioneers had turned their telescopes to it before Peiresc.

In one of the most detailed astronomical images ever produced, NASA/ESA’s Hubble Space Telescope captured an unprecedented look at the Orion Nebula. … This extensive study took 105 Hubble orbits to complete. All imaging instruments aboard the telescope were used simultaneously to study Orion. Source: Wikimedia Commons

Peiresc, like Galileo, realised that the moons of Jupiter could be used as a clock to determine longitude and began an observation programme of the moons, viewing them every single day that the weather conditions permitted, well into 1612. Having compiled tables of his observations he sent one of his own protégés Jean Lombard, about whom little is known, equipped with suitable instruments on a tour of the Mediterranean. Lombard observed the satellites at Marseille in November 1611 and then proceeded to Malta, Cyprus and to Tripoli observing as he went, until May 1612. Meanwhile, Peiresc made parallel observation in Aix and Paris, he hoped by comparing the time differences in the two sets of observations to be able to accurately determine the longitude differences. Unfortunately, the observations proved to be not accurate enough for the purpose and the world would have to wait for Giovanni Domenico Cassini (1625–1712) to become the first to successfully utilise this method of determining longitude. Peiresc’s own observation were, however, the longest continuous series of observations of the Jupiter moons made in the seventeenth century and displayed a high level of accuracy even when compared with this of Galileo.

I mentioned, above, Peiresc’s Provencal astronomical observing group. Peiresc employed/sponsored young astronomers to help him with his observation programmes, supplying them with instruments and instructions on how to use them. This group included such notable, future astronomers, as Jean-Baptiste Morin (1583–1556),

Jean-Baptiste Morin Source: Wikimedia Commons

Ismaël Boulliau (1605–1694),

Ismaël Boulliau Source: Wikimedia Commons

and Pierre Gassendi (1592–1655). Peiresc’s patronage extended well beyond this. Gassendi had held the chair of philosophy at the University of Aix-en-Provence since 1617 but in 1623 the university was taken over by the Jesuits and Gassendi was replaced by a Jesuit and became unemployed.

Portrait of Pierre Gassendi by Louis-Édouard Rioult Source: Wikimedia Commons

From then until he again found regular employment in 1634, Peiresc provided him with a home base in his own house and financed his travels and research. Similarly, Peiresc, having introduced Mersenne to Parisian intellectual circles in 1619, continued to support him financially, Mersenne as a Minim friar had no income, supplying him with instruments and financing his publications. 

Marin Mersenne Source: Wikimedia Commons

Patronage played a central role in Peiresc’s next venture into astronomy and another attempt to solve the longitude problem. There has been much talk in recent decades about so-called citizen science, in which members of the public are invited to participate in widespread scientific activities. Annual counts of the birds in one’s garden is a simple example of this. Citizen science is mostly presented as a modern phenomenon, but there are examples from the nineteenth century. Peiresc had already launched a variation on citizen science in the seventeenth century.

In order to determine longitude Peiresc further developed a method that had been in use since antiquity. Two astronomers situated in different location would observe a lunar or solar eclipse and then by comparing their observations they could determine the local time difference between their observations and thus the longitude difference between the locations. By the seventeenth century predicting eclipses had become a fairly accurate science and Peiresc thought that if he could organise and coordinated a world spanning network of observers to accurately observe and record an eclipse, he could then calculate a world spanning network of longitude measurements. The idea was good in theory but failed in practice.

Most of Peiresc’s team of observers were amateurs–missionaries, diplomats, traders, travellers–whom he supplied with astronomical instruments and written instructions on how to use them, even paying travelling expenses, where necessary. Peiresc organised mass observations for lunar eclipses in 1628, 1634, and 1635 and a solar eclipse in 1633. Unfortunately, many of his observers proved to be incompetent and the results of their observations were too inaccurate to be usable. One positive result was that Peiresc was able to correct the value for the length of the Mediterranean. Before one is too hard on Peiresc’s amateur observers, one should remember that in the middle of the eighteenth century the world’s professional astronomical community basically failed in their attempt to use the transits of Venus to determine the astronomical unit, despite being equipped with much better instruments and telescopes.

Although, Peiresc’s various astronomical activities and their results were known throughout Europe by word of mouth through his various colleagues and his correspondence, he never published any of his work. Quite why, is not really known although there are speculations.

Peiresc was a high ranking and highly influential Catholic and he applied that influence in attempts to change the Church’s treatment of astronomers he saw as being persecuted. He interceded on behalf Tommaso Campanella (1568–1639), actively supporting him when he fled to France in 1634.

Tommaso Campanella portrait by Francesco Cozza Source: Wikimedia Commons

More famously he personally interceded with the Church on behalf of Galileo, without any great success.

Nicolas-Claude Fabri de Peiresc’s career is, like that of his friend Mersenne, a good illustration that the evolution of science is a product of widespread cooperation of a community of practitioners and not the result of the genial discoveries of a handful of big names, as it is unfortunately too often presented. Morin, Boulliau, Gassendi and Mersenne, who all made serious contributions to the evolution of science in the seventeenth century, did so with the encouragement, guidance, and very active support of Peiresc.


Filed under History of Astronomy, History of Navigation, History of science, Renaissance Science, The Paris Provencal Connection

Internet Superstar, who are you, what do you think you are?

He’s back!

After his stupendously, mind-bogglingly, world shattering success rabbiting on about the history of astronomy on the History for Atheists YouTube channel, he can now be heard going on and on and on and on and on and on…  about the history of astronomy from Babylon to Galileo Galilei on the monumental, prodigious, phenomenal Subject to Change podcast, moderated by sensational Russell Hogg and available on so many different Internet channels you’ll need a week to decide where to listen. 


Filed under Autobiographical, History of Astrology, History of Astronomy