Category Archives: History of Mathematics

Renaissance science – XXVII

Early on in this series I mentioned that a lot of the scientific developments that took place during the Renaissance were the result of practical developments entering the excessively theoretical world of the university disciplines. This was very much the case in the mathematical sciences, where the standard English expression for the Renaissance mathematicus is mathematical practitioner. In this practical world, areas that we would now regard as separate disciples were intertwined is a complex that the mathematical practitioners viewed as one discipline with various aspects, this involved astronomy, cartography, navigation, trigonometry, as well as instrument and globe making. I have already dealt with trigonometry, cartography and astronomy and will here turn my attention to navigation, which very much involved the other areas in that list.

The so-called Age of Discovery or Age of Exploration, that is when Europeans started crossing the oceans and discovering other lands and other cultures, coincides roughly with the Renaissance and this was, of course the main driving force behind the developments in navigation during this period. Before we look at those developments, I want to devote a couple of lines to the terms Age of Discovery and Age of Exploration. Both of them imply some sort of European superiority, “you didn’t exist until we discovered you” or “your lands were unknown until we explored them.” The populations of non-European countries and continents were not sitting around waiting for their lands and cultures to be discovered by the Europeans. In fact, that discovery very often turned out to be highly negative for the discovered. The explorers and discoverers were not the fearless, visionary heroes that we tend to get presented with in our schools, but ruthless, often brutal chancers, who were out to make a profit at whatever cost.  This being the case the more modern Contact Period, whilst blandly neutral, is preferred to describe this period of world history.

As far as can be determined, with the notable exception of the Vikings, sailing in the Atlantic was restricted to coastal sailing before the Late Middle Ages. Coastal sailing included things such as crossing the English Channel, which, archaeological evidence suggests, was done on a regular basis since at least the Neolithic if not even earlier. I’m not going to even try to deal with the discussions about how the Vikings possibly navigated. Of course, in other areas of the world, crossing large stretches of open water had become common place, whilst the European seamen still clung to their coast lines. Most notable are the island peoples of the Pacific, who were undertaking long journeys across the ocean already in the first millennium BCE. Arab and Chinese seamen were also sailing direct routes across the Indian Ocean, rather than hugging the coastline, during the medieval period. It should be noted that European exploited the navigation skills developed by these other cultures as they began to take up contact with the other part of the world. Vasco da Gamma (c. 1460–1524) used unidentified local navigators to guide his ships the first time he crossed the Indian Ocean from Africa to India. On his first voyage of exploration of the Pacific Ocean from 1768 to 1771, James Cook (1728–1779) used the services of the of the Polynesian navigator, Tupaia (c. 1725–1770), who even drew a chart, in cooperation with Cook, Joseph Banks, and several of Cooks officer, of his knowledge of the Pacific Ocean. 

Tupaia’s map, c. 1769 Source: Wikimedia Commons

There were two major developments in European navigation during the High Middle Ages, the use of the magnetic compass and the advent of the Portolan chart. The Chinese began to use the magnetic properties of loadstone, the mineral magnetite, for divination sometime in the second century BCE. Out of this they developed the compass needle over several centuries. It should be noted that for the Chinese, the compass points South and not North. The earliest Chinese mention of the use of a compass for navigation on land by the military is before 1044 CE and in maritime navigation in 1117 CE.

Diagram of a Ming Dynasty (1368–1644) mariner’s compass Source: Wikimedia Commons

Alexander Neckam (1157–1219) reported the use of the compass for maritime navigation in the English Channel in his manuscripts De untensilibus and De naturis rerum, written between 1187 and 1202.

The sailors, moreover, as they sail over the sea, when in cloudy whether they can no longer profit by the light of the sun, or when the world is wrapped up in the darkness of the shades of night, and they are ignorant to what point of the compass their ship’s course is directed, they touch the magnet with a needle, which (the needle) is whirled round in a circle until, when its motion ceases, its point looks direct to the north.

This and other references to the compass suggest that it use was well known in Europe by this time.

A drawing of a compass in a mid 14th-century copy of Epistola de magnete of Peter Peregrinus. Source: Wikimedia Commons

The earliest reference to maritime navigation with a compass in the Muslim world in in the Persian text Jawāmi ul-Hikāyāt wa Lawāmi’ ul-Riwāyāt (Collections of Stories and Illustrations of Histories) written by Sadīd ud-Dīn Muhammad Ibn Muhammad ‘Aufī Bukhārī (1171-1242) in 1232. There is still no certainty as to whether there was a knowledge transfer from China to Europe, either direct or via the Islamic Empire, or independent multiple discovery. Magnetism and the magnetic compass went through a four-hundred-year period of investigation and discovery until William Gilbert (1544–1603) published his De magnete in 1600. 

De Magnete, title page of 1628 edition Source: Wikimedia Commons

The earliest compasses used for navigation were in the form of a magnetic needle floating in a bowl of water. These were later replaced with dry mounted magnetic needles. The first discovery was the fact that the compass needle doesn’t actually point at the North Pole, the difference is called magnetic variation or magnetic declination. The Chinese knew of magnetic declination in the seventh century. In Europe the discovery is attributed to Georg Hartmann (1489–1564), who describes it in an unpublished letter to Duke Albrecht of Prussia. However, Georg von Peuerbach (1423–1461) had already built a portable sundial on which the declination for Vienna is marked on the compass.

NIMA Magnetic Variation Map 2000 Source: Wikimedia Commons

There followed the discovery that magnetic declination varies from place to place. Later in the seventeenth century it was also discovered that declination also varies over time. We now know that the Earth’s magnetic pole wanders, but it was first Gilbert, who suggested that the Earth is a large magnet with poles. The next discovery was magnetic dip or magnetic inclination. This describes the fact that a compass needle does not sit parallel to the ground but points up or down following the lines of magnetic field. The discovery of magnetic inclination is also attributed to Georg Hartmann. The sixteenth century English, seaman Robert Norman rediscovered it and described how to measure it in his The Newe Attractive (1581) His work heavily influenced Gilbert. 

Illustration of magnetic dip from Norman’s book, The Newe Attractive Source: Wikimedia Commons

The Portolan chart, the earliest known sea chart, emerged in the Mediterranean in the late thirteenth century, not long after the compass, with which it is closely associated, appeared in Europe. The oldest surviving Portolan, the Carta Pisana is a map of the Mediterranean, the Black Sea and part of the Atlantic coast.

Source: Wikimedia Commons

The origins of the Portolan chart remain something of a mystery, as they are very sophisticated artifacts that appear to display no historical evolution. A Portolan has a very accurate presentation of the coastlines with the locations of the major harbours and town on the coast. Otherwise, they have no details further inland, indicating that they were designed for use in coastal sailing. A distinctive feature of Portolans is their wind roses or compass roses located at various points on the charts. These are points with lines radiating outwards in the sixteen headings, on later charts thirty-two, of the mariner’s compass.

Central wind rose on the Carta Pisana

Portolan charts have no latitude or longitude lines and are on the so-called plane chart projection, which treats the area being mapped as flat, ignoring the curvature of the Earth. This is alright for comparatively small areas, such as the Mediterranean, but leads to serious distortion, when applied to larger areas.

During the Contact Period, Portolan charts were extended to include the west coast of Africa, as the Portuguese explorers worked their way down it. Later, the first charts of the Americas were also drawn in the same way. Portolan style charts remained popular down to the eighteenth century.

Portolan chart of Central America c. 1585-1595 Source:

A central problem with Portolan charts over larger areas is that on a globe constant compass bearings are not straight lines. The solution to the problem was found by the Portuguese cosmographer Pedro Nunes (1502–1578) and published in his Tratado em defensam da carta de marear (Treatise Defending the Sea Chart), (1537).

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

The line is a spiral known as a loxodrome or rhumb lines. Nunes problem was that he didn’t know how to reproduce his loxodromes on a flat map.

Image of a loxodrome, or rhumb line, spiraling towards the North Pole Source: Wikimedia Commons

The solution to the problem was provided by the map maker Gerard Mercator (1512–1594), when he developed the so-called Mercator projection, which he published as a world map, Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendate Accommodata (New and more complete representation of the terrestrial globe properly adapted for use in navigation) in 1569.

Source: Wikimedia Commons
The 1569 Mercator world map Source: Wikimedia Commons.

On the Mercator projection lines of constant compass bearing, loxodromes, are straight lines. This however comes at a price. In order to achieve the required navigational advantage, the lines of latitude on the map get further apart as one moves away from the centre of projection. This leads to an area distortion that increases the further north or south on goes from the equator. This means that Greenland, slightly more than two million square kilometres, appear lager than Africa, over thirty million square kilometres.

Mercator did not publish an explanation of the mathematics used to produce his projection, so initially others could reproduce it. In the late sixteenth century three English mathematicians John Dee (1527–c. 1608), Thomas Harriot (c. 1560–1621), and Edward Wright (1561–1615) all individually worked out the mathematics of the Mercator projection. Although Dee and Harriot both used this knowledge and taught it to others in their respective functions as mathematical advisors to the Muscovy Trading Company and Sir Walter Raleigh, only Wright published the solution in his Certaine Errors in Navigation, arising either of the Ordinarie Erroneous Making or Vsing of the Sea Chart, Compasse, Crosse Staffe, and Tables of Declination of the Sunne, and Fixed Starres Detected and Corrected. (The Voyage of the Right Ho. George Earle of Cumberl. to the Azores, &c.) published in London in 1599. A second edition with a different, even longer, title was published in the same year. Further editions were published in 1610 and 1657. 

Source: Wikimedia Commons
Wright explained the Mercator projection with the analogy of a sphere being inflated like a bladder inside a hollow cylinder. The sphere is expanded uniformly, so that the meridians lengthen in the same proportion as the parallels, until each point of the expanding spherical surface comes into contact with the inside of the cylinder. This process preserves the local shape and angles of features on the surface of the original globe, at the expense of parts of the globe with different latitudes becoming expanded by different amounts. The cylinder is then opened out into a two-dimensional rectangle. The projection is a boon to navigators as rhumb lines are depicted as straight lines. Source: Wikimedia Commons

His mathematical solution for the Mercator projection had been published previously with his permission and acknowledgement by Thomas Blundeville (c. 1522–c. 1606) in his Exercises (1594) and by William Barlow (died 1625) in his The Navigator’s Supply (1597). However, Jodocus Hondius (1563–1612) published maps using Wright’s work without acknowledgement in Amsterdam in 1597, which provoked Wright to publish his Certaine Errors. Despite its availability, the uptake on the Mercator projection was actually very slow and it didn’t really come into widespread use until the eighteenth century.

Wright’s “Chart of the World on Mercator’s Projection” (c. 1599), otherwise known as the Wright–Molyneux map because it was based on the globe of Emery Molyneux (died 1598) Source: Wikimedia Commons

Following the cartographical trail, we have over sprung a lot of developments in navigation to which we will return in the next episode. 


Filed under History of Cartography, History of Mathematics, History of Navigation, 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. 

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

Renaissance Science – XXV

It is generally acknowledged that the mathematisation of science was a central factor in the so-called scientific revolution. When I first came to the history of science there was widespread agreement that this mathematisation took place because of a change in the underlaying philosophy of science from Aristotelian to Platonic philosophy. However, as we saw in the last episode of this series, the renaissance in Platonic philosophy was largely of the Neoplatonic mystical philosophy rather than the Pythagorean, mathematical Platonic philosophy, the Plato of “Let no one ignorant of geometry enter here” inscribed over the entrance to The Academy. This is not to say that Plato’s favouring of mathematics did not have an influence during the Renaissance, but that influence was rather minor and not crucial or pivotal, as earlier propagated.

It shouldn’t need emphasising, as I’ve said it many times in the past, but Galileo’s infamous, Philosophy is written in this grand book, which stands continually open before our eyes (I say the ‘Universe’), but can not be understood without first learning to comprehend the language and know the characters as it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible to humanly understand a word; without these one is wandering in a dark labyrinth, is not the origin of the mathematisation, as is falsely claimed by far too many, who should know better. One can already find the same sentiment in the Middle Ages, for example in Islam, in the work of Ibn al-Haytham (c. 965–c. 1040) or in Europe in the writings of both Robert Grosseteste (c. 1168–1253) and Roger Bacon, (c. 1219–c. 1292) although in the Middle Ages, outside of optics and astronomy, it remained more hypothetical than actually practiced. We find the same mathematical gospel preached in the sixteenth century by several scholars, most notably Christoph Clavius (1538–1612).

As almost always in history, it is simply wrong to look for a simple mono-casual explanation for any development. There were multiple driving forces behind the mathematisation. As we have already seen in various earlier episodes, the growing use and dominance of mathematics was driving by various of the practical mathematical disciplines during the Renaissance. 

The developments in cartography, surveying, and navigation (which I haven’t dealt with yet) all drove an increased role for both geometry and trigonometry. The renaissance of astrology also served the same function. The commercial revolution, the introduction of banking, and the introduction of double entry bookkeeping all drove the introduction and development of the Hindu-Arabic number system and algebra, which in turn would lead to the development of analytical mathematics in the seventeenth century. The development of astro-medicine or iatromathematics led to a change in the status of mathematic on the universities and the demand for commercial arithmetic led to the establishment of the abbacus or reckoning schools. The Renaissance artist-engineers with their development of linear perspective and their cult of machine design, together with the new developments in architecture were all driving forces in the development of geometry. All of these developments both separately and together led to a major increase in the status of the mathematical sciences and their dissemination throughout Europe. 

This didn’t all happen overnight but was a gradual process spread over a couple of centuries. However, by the early seventeenth century and what is generally regarded as the start of the scientific revolution the status and spread of mathematics was considerably different, in a positive sense, to what it had been at the end of the fourteenth century. Mathematics was now very much an established part of the scholarly spectrum. 

There was, however, another force driving the development and spread of mathematics and that was surprisingly the, on literature focused, original Renaissance humanists in Northern Italy. In and of itself, the original Renaissance humanists did not measure mathematics an especially important role in their intellectual cosmos. So how did the humanists become a driving force for the development of mathematics? The answer lies in their obsession with all and any Greek or Latin manuscripts from antiquity and also with the attitude to mathematics of their ancient role models. 

Cicero admired Archimedes, so Petrarch admired Archimedes and other humanists followed his example. In his Institutio Oratoria Quintilian was quite enthusiastic about mathematics in the training of the orator. However, both Cicero and Quintilian had reservations about how too intense an involvement with mathematics distracts one from the active life. This meant that the Renaissance humanists were, on the whole, rather ambivalent towards mathematics. They considered it was part of the education of a scholar, so that they could converse reasonably intelligently about mathematics in general, but anything approaching a deep knowledge of the subject was by and large frowned upon. After all, socially, mathematici were viewed as craftsmen and not scholars.

This attitude stood in contradiction to their manuscript collecting habits. On their journeys to the cloister libraries and to Byzantium, the humanists swept up everything they could find in Latin and/or Greek that was from antiquity. This meant that the manuscript collections in the newly founded humanist libraries also contained manuscripts from the mathematical disciplines. A good example is the manuscript of Ptolemaeus’ Geographia found in Constantinople and translated into Latin by Jacobus Angelus for the first time in 1406. The manuscripts were now there, and scholars began to engage with them leading to a true mathematical renaissance of the leading Greek mathematicians. 

We have already seen, in earlier episodes, the impact that the works of Ptolemaeus, Hero of Alexander, and Vitruvius had in the Renaissance, now I’m going to concentrate on three mathematicians Euclid, Archimedes, and Apollonius of Perga, starting with Archimedes. 

The works of Archimedes had already been translated from Greek into Latin by the Flemish translator Willem van Moerbeke (1215–1286) in the thirteenth century.

Archimedes Greek manuscript

He also translated texts by Hero. Although, he was an excellent translator, he was not a mathematician, so his translations were somewhat difficult to comprehend. Archimedes was to a large extent ignored by the universities in the Middle Ages. In 1530, Jacobus Cremonensis (c. 1400–c. 1454) (birth name Jacopo da San Cassiano), a humanist and mathematician, translated, probably at request of the Pope, Nicholas V (1397–1455), a Greek manuscript of the works of Archimedes into Latin. He was also commissioned to correct George of Trebizond’s defective translation of Ptolemaeus’ Mathēmatikē Syntaxis. It is thought that the original Greek manuscript was lent or given to Basilios Bessarion (1403–1472) and has subsequently disappeared.

Bessarion had not only the largest humanist library but also the library with the highest number of mathematical manuscripts. Many of Bessarion’s manuscripts were collected by Regiomontanus (1436–1476) during the four to five years (1461–c. 1465) that he was part of Bessarion’s household.

Basilios Bessarion Justus van Gent and Pedro Berruguete Source: Wikimedia Commons

When Regiomontanus moved to Nürnberg in 1471 he brought a large collection of mathematical, astronomical, and astrological manuscripts with him, including the Cremonenius Latin Archimedes and several manuscripts of Euclid’s Elements, that he intended to print and publish in the printing office that he set up there. Unfortunately, he died before he really got going and had only published nine texts including his catalogue of future intended publications that also listed the Cremonenius Latin Archimedes. 

The invention of moving type book printing was, of course, a major game changer. In 1482, Erhard Ratdolt (1447–1522) published the first printed edition of The Elements of Euclid from one of Regiomontanus’ manuscripts of the Latin translation from Arabic by Campanus of Novara (c. 1220–1296).

A page with marginalia from the first printed edition of Euclid’s Elements, printed by Erhard Ratdolt in 1482
Folger Shakespeare Library Digital Image Collection
Source: Wikimedia Commons

In 1505, a Latin translation from the Greek by Bartolomeo Zamberti (c. 1473–after 1543) was published in Venice in 1505, because Zamberti regarded the Campanus translation as defective. The first Greek edition, edited by Simon Grynaeus (1493–1541) was published by Jacob Herwegens in Basel in 1533.

Simon Grynaeus Source: Wikimedia Commons
Editio princeps of the Greek text of Euclid. Source

Numerous editions followed in Greek and/or Latin. The first modern language edition, in Italian, translated by the mathematician Niccolò Fontana Tartaglia (1499/1500–1557) was published in 1543.

Tartaglia Euclid Source

Other editions in German, French and Dutch appeared over the years and the first English edition, translated by Henry Billingsley (died 1606) with a preface by John Dee (1527–c. 1608) was published in 1570.

Title page of Sir Henry Billingsley’s first English version of Euclid’s Elements Source Wikimedia Commons

In 1574, Christoph Clavius (1538–1612) published the first of many editions of his revised and modernised Elements, to be used in his newly inaugurated mathematics programme in Catholic schools, colleges, and universities. It became the standard version of Euclid throughout Europe in the seventeenth century. In 1607, Matteo Ricci (1552–1610) and Xu Guanqui (1562–1633) published their Chinese translation of the first six books of Clavius’ Elements.

Xu Guangqi with Matteo Ricci (left) From Athanasius Kircher’s China Illustrata, 1667 Source: Wikimedia Commons

From being a medieval university textbook of which only the first six of the thirteen books were studied if at all, The Elements was now a major mathematical text. 

Unlike his Euclid manuscript, Regiomontanus’ Latin Archimedes manuscript had to wait until the middle of the sixteenth century to find an editor and publisher. In 1544, Ioannes Heruagius (Johannes Herwagen) (1497–1558) published a bilingual, Latin and Greek, edition of the works of Archimedes, edited by the Nürnberger scholar Thomas Venatorius (Geschauf) (1488–1551).

Thomas Venatorius Source

The Latin was the Cremonenius manuscript that Regiomontanus had brought to Nürnberg, and the Greek was a manuscript that Willibald Pirckheimer (1470–1530) had acquired in Rome.

Venatori Archimedes Source

Around the same time Tartaglia published partial editions of the works of Archimedes both in Italian and Latin translation. We will follow the publication history of Archimedes shortly, but first we need to go back to see what happened to The Conics of Apollonius, which became a highly influential text in the seventeenth century.

Although, The Conics was known to the Arabs, no translation of it appears to have been made into Latin during the twelfth-century scientific Renaissance. Giovanni-Battista Memmo (c. 1466–1536) produced a Latin translation of the first four of the six books of The Conics, which was published posthumously in Venice in 1537. Although regarded as defective this remained the only edition until the latter part of the century.

Memmo Apollonius Conics Source: Wikimedia Commons

We now enter the high point of the Renaissance reception of both Archimedes and Apollonius in the work of the mathematician and astronomer Francesco Maurolico (1494–1575) and the physician Federico Commandino (1509-1575). Maurolico spent a large part of his life improving the editions of a wide range of Greek mathematical works.

L0006455 Portrait of F. Maurolico by Bovis after Caravaggio Credit: Wellcome Library, London, via Wikimedia Commons

Unfortunately, he had problems finding sponsors and/or publishers for his work. His heavily edited and corrected volume of the works of Archimedes first appeared posthumously in Palermo in 1585. His definitive Latin edition of The Conics, with reconstructions of the fifth and sixth books, completed in 1547, was first published in 1654.

Maurolico corresponded with Christoph Clavius, who had visited him in Sicily in 1574, when the observed an annular solar eclipse together, and with Federico Commandino, although the two never met.

Federico Commandino produced and published a whole series of Greek mathematical works, which became something like standard editions.

Source: Wikimedia Commons

His edition of the works of Archimedes appeared in 1565 and his Apollonius translation in 1566.

Two of Commandino’s disciples were Guidobaldo del Monte (1545–1607) and Bernardino Baldi (1553–1617). 

Baldi wrote a history of mathematics the Cronica dei Matematici, which was published in Urbino in 1707. This was a brief summary of his much bigger Vite de’ mathematici, a two-thousand-page manuscript that was never published.

Bernadino Baldi Source: Wikimedia Commons
Source: Wikimedia Commons

Guidobaldo del Monte, an aristocrat, mathematician, philosopher, and astronomer

Guidobaldo del Monte Source: Wikimedia Commons

became a strong promoter of Commandino’s work and in particular the works of Archimedes, which informed his own work in mechanics. 

In the midst of that darkness Federico Commandino shone like the sun, for his learning he not only restored the lost heritage of mathematics but actually increased and enhanced it … In him seem to have lived again Archytas, Diophantus, Theodosius, Ptolemy, Apollonius, Serenus, Pappus and even Archimedes himself.

Guidobaldo. Liber Mechanicorum, Pesaro 1577, Preface
Source: Wikimedia Commons

When the young Galileo wrote his first essay on the hydrostatic balance, his theory how Archimedes actually detected the substitution of silver for gold in the crown made for King Hiero of Syracuse, he sent it to Guidobaldo to try and win his support and patronage. Guidobaldo was very impressed and got his brother Cardinal Francesco Maria del Monte (1549–1627), the de’ Medici family cardinal, to recommend Galileo to Ferinando I de’ Medici, Grand Duke of Tuscany, (1549–1609) for the position of professor of mathematics at Pisa University. Galileo worked together with Guidobaldo on various projects and for Galileo, who rejected Aristotle, Archimedes became his philosophical role model, who he often praised in his works. 

Galileo was by no means the only seventeenth century scientist to take Archimedes as his role model in pursuing a mathematical physics, for example Kepler used a modified form of Archimedes’ method of exhaustion to determine the volume of barrels, a first step to the development of integral calculus. The all pervasiveness of Archimedes in the seventeenth century is wonderfully illustrated at the end of the century by Sir William Temple, Jonathan Swift’s employer, during the so-called battle of the Ancients and Moderns. In one of his essays, Temple an ardent supporter of the superiority of the ancients over the moderns, asked if John Wilkins was the seventeenth century Archimedes, a rhetorical question with a definitively negative answer. 

During the Middle Ages Euclid was the only major Greek mathematician taught at the European universities and that only at a very low level. By the seventeenth century Euclid had been fully restored as a serious mathematical text and the works of both Archimedes and Apollonius had entered the intellectual mainstream and all three texts along with other restored Greek texts such as the Mathematical Collection of Pappus, also published by Commandino and the Arithmetica of Diophantus, another manuscript brought to Nürnberg by Regiomontanus and worked on by numerous mathematicians, became influential in development of the new sciences.  


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

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.

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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

Musical, mathematical Minim, Marin Mersenne 

In the seventeenth century, Marin Mersenne (1588–1648) was a very central and highly influential figure in the European intellectual and scientific communities; a man, who almost literally knew everybody and was known by everybody in those communities. Today, in the big names, big events, popular versions of the history of science he remains only known to specialist historian of science and also mathematicians, who have heard of Mersenne Primes, although most of those mathematicians probably have no idea, who this Mersenne guy actually was. So, who was Marin Mersenne and why does he deserve to be better known than he is?

Marin Mersenne Source: Wikimedia Commons

Mersenne was born 8 September 1588, the son of Julien Mersenne and his wife Jeanne, simple peasants, in Moulière near Oizé, a small commune in the Pays de la Loire in North-Western France. He was first educated at the at the nearby College du Mans and then from 1604 to 1609 at the newly founded Jesuit Collège Henri-IV de La Flèche. The latter is important as in La Flèche he would have received the mathematical programme created by Christoph Clavius for the Jesuit schools and colleges, the best mathematical education available in Europe at the time. A fellow student at La Flèche was René Descartes (1596–1650) with whom he would become later in life close friends.

René Descartes at work Source: Wikimedia Commons

However, it is unlikely that they became friends then as Mersenne was eight years older. Leaving La Flèche he continued his education in Greek, Hebrew, and theology at the Collège Royal and the Sorbonne in Paris. In 1611 he became a Minim friar and a year later was ordained as a priest. The Minims are a mendicant order founded in Italy in the fifteenth century. From 1614 to 1618 he taught philosophy and theology at Nevers but was recalled to Paris in 1619 to the newly established house on the Place Royal (now Place des Vosges), where he remained, apart from travels through France, to Holland, and to Italy, until his death. 

View map of an area of Paris near Place Royale, now Place des Vosges, showing the Minim convent where Mersenne lived and the Rue des Minimes, not far from the Bastille, undated, but before 1789 ( Source: Linda Hall Library

In Paris he was introduced to the intellectual elite by Nicolas-Claude Fabri de Pereisc (1580–1637)–wealthy astronomer, antiquarian, and patron of science–whom he had got to know in 1616. 

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

Settled in Paris, Mersenne began a career as a prolific author, both editing and publishing new editions of classical works and producing original volumes. In the 1620s his emphasis was on promoting and defending the Thomist, Aristotelian philosophy and theology in which he’d been educated. In his first book, Questiones celeberrimae in Genesim (1623), 

he attacked those he saw as opponents of the true Catholic religion, Platonist, cabbalistic and hermetic authors such as Telesio, Pomponazzi, Bruno and Robert Fludd. His second book, L’impiété des déistes, athées, et libertins de ce temps (1624), continued his attacks on the propagators of magic and the occult. His third book, La Vérité des sciences (1625), attacks alchemists and sceptics and includes a compendium of texts over ancient and recent achievements in the mathematical sciences that he saw as in conformity with his Christian belief. The latter drew the attention of Pierre Gassendi (1592–1655), who became his closest friend. I shall return to their joint activities in Paris later but now turn to Mersenne’s own direct scientific contributions, which began to replace the earlier concentration on theology and philosophy.

Pierre Gassendi Source: Wikimedia Commons

Mersenne’s scientific interests lay in mathematics and in particular what Aristotle, who was not a fan of mathematics, claiming it did not apply to the real world, called the mixed sciences or mixed mathematics i.e., astronomy, optics, statics, etc. Here he compiled to collections of treatises on mixed mathematics, his Synopsis Mathematica (1626) and Universae geometriae synopsis (1644). In his Traité de l’Harmonie Universelle (1627), to which we will return, Mersenne gives a general introduction to his concept of the mathematical disciplines:

Geometry looks at continuous quantity, pure and deprived from matter and from everything which falls upon the senses; arithmetic contemplates discrete quantities, i.e. numbers; music concerns har- monic numbers, i.e. those numbers which are useful to the sound; cosmography contemplates the continuous quantity of the whole world; optics looks at it jointly with light rays; chronology talks about successive continuous quantity, i.e. past time; and mechanics concerns that quantity which is useful to machines, to the making of instruments and to anything that belongs to our works. Some also adds judiciary astrology. However, proofs of this discipline are borrowed either from astronomy (that I have comprised under cosmology) or from other sciences. 

In optics he addressed the problem of spherical aberration in lenses and mirrors and suggested a series of twin mirror reflecting telescopes, which remained purely hypothetical and were never realised.

Source: Fred Watson, “Stargazer: The Life and Times of the Telescope”, Da Capo Press, 2004, p. 115

This is because they were heavily and falsely criticised by Descartes, who didn’t really understand them. It was Mersenne, who pushed Descartes to his solution of the refraction problem and the discovery of the sine law. He wrote three books on optics, De Natura lucis (1623); Opticae (1644); L’Optique et la catoptrique (1651). Although his theoretical reflecting telescopes were published in his Harmonie universelle (1636), see below.

Mersenne also wrote and published collections of essays on other areas of mixed mathematics, mechanics, pneumatics, hydro- statics, navigation, and weights and measures, Cogitata physico-mathematica (1644); Novarum observationum physico- mathematicarum tomus III (1647). 

Mersenne dabbled a bit in mathematics itself but unlike many of his friends did not contribute much to pure mathematics except from the Mersenne prime numbers those which can be written in the form Mn = 2n − 1 for some integer n. This was his contribution to a long search by mathematicians for some form of law that consistently generates prime numbers. Mersenne’s law whilst generating some primes doesn’t consistently generate primes but it has been developed into its own small branch of mathematics. 

It was, however, in the field of music, as the title quoted above would suggest, which had been considered as a branch of mathematics in the quadrivium since antiquity, and acoustics that Mersenne made his biggest contribution. This has led to him being labelled the “father of acoustics”, a label that long term readers of this blog will know that I reject, but one that does to some extent encapsulate his foundational contributions to the discipline. He wrote and published five books on the subject over a period of twenty years–Traité de l’harmonie universelle (1627); Questions harmoniques (1634); Les preludes de l’harmonie universelle (1634); Harmonie universelle (1636); Harmonicorum libri XII (1648)–of which his monumental (800 page) Harmonie universelle was the most important and most influential.

Title page of Harmonie universelle Source: Wikimedia Commons

In this work Mersenne covers the full spectrum including the nature of sounds, movements, consonance, dissonance, genres, modes of composition, voice, singing, and all kinds of harmonic instruments. Of note is the fact that he looks at the articulation of sound by the human voice and not just the tones produced by instruments. He also twice tried to determine the speed of sound. The first time directly by measuring the elapse of time between observing the muzzle flash of a cannon and hearing the sound of the shot being fired. The value he determined 448 m/s was higher than the actual value of 342 m/s. In the second attempt, recorded in the Harmonie universelle (1636), he measured the time for the sound to echo back off a wall at a predetermined distance and recorded the value of 316 m/s. So, despite the primitive form of his experiment his values were certainly in the right range. 

Mersenne also determined the correct formular for determining the frequency of a vibrating string, something that Galileo’s father Vincenzo (1520–1591) had worked on. This is now known as Mersenne’s Law and states that the frequency is inversely proportional to the length of the string, proportional to the square root of the stretching force, and inversely proportional to the square root of the mass per unit length.

The formula for the lowest frequency is f=\frac{1}{2L}\sqrt{\frac{F}{\mu}},

where f is the frequency [Hz], L is the length [m], F is the force [N] and μ is the mass per unit length [kg/m].

Source: Wikipedia

Vincenzo Galileo was also involved in a major debate about the correct size of the intervals on the musical scale, which was rumbling on in the late sixteenth and early seventeenth centuries. It was once again Mersenne, who produced the solution that we still use today.

Although Mersenne is certainly credited and honoured by acoustic researchers and music theorists for his discoveries in these areas, perhaps his most important contribution to the development of the sciences in the seventeenth century was as a networker and science communicator in a time when scientific journals didn’t exist yet. 

Together with Gassendi he began to hold weekly meetings in his humble cell with other natural philosophers, mathematicians, and other intellectuals in Paris. Sometime after 1633 these meetings became weekly and took place in rotation in the houses of the participants and acquired the name Academia Parisiensis. The list of participants reads like an intellectual who’s who of seventeenth century Europe and included René Descartes, Étienne Pascal and his son Blaise, Gilles de Roberville, Nicolas-Claude Fabri de Pereisc, Pierre de Fermat, Claude Mydorge, the English contigent, Thomas Hobbes, Kenhelm Digby, and the Cavendishes, and for those not living in or near Paris such as Isaac Beeckman, Jan Baptist van Helmont, Constantijn Huygens and his son Christiaan, and not least Galileo Galilei by correspondence. When he died approximately six hundred letters were found in his cell from seventy-nine different correspondents. In total 193 scholars and literati have been identified as participants. Here it should be noted that although he tended to reject the new emerging sciences in his earlier defence of Thomist philosophy, he now embraced it as compatible with his teology and began to promote it.

This academy filled a similar function to the Gresham College group and Hartlib Circle in England, as well as other groups in other lands, as precursors to the more formal scientific academies such as the Académie des sciences in Paris and the Royal Society in London. There is evidence that Jean-Baptist Colbert (1619–1683), the French Minister of State, modelled his Académie des sciences on the Academia Parisiensis. Like its formal successors the Academia Parisiensis served as a forum for scholars to exchange views and theories and discuss each other’s work. Mersenne’s aim in establishing this forum was to stimulate cooperation between the participants believing science to be best followed as a collective enterprise.

Mersenne’s role was not restricted to that of convener, but he functioned as a sort of agent provocateur deliberately stimulating participants to take up research programmes that he inaugurated. For example, he brought Torricelli’s primitive barometer to Paris and introduced it to the Pascals. It is thought that he initiated the idea to send Blaise Pascal’s brother-in-law up the Puy de Dôme to measure the decreasing atmospheric pressure.

Blaise Pascal, unknown; a copy of the painting of François II Quesnel, which was made for Gérard Edelinck in 1691. Source: Wikimedia Commons

Although they never met and only corresponded, he introduced Christiaan Huygens to the concept of using a pendulum to measure time, leading to Huygens’ invention of the pendulum clock.

Portrait of Christiaan Huygens (1629-1695) C.Netscher / 1671 Source: Wikimedia Commons

It was Mersenne, who brought the still very young Blaise Pascal together with René Descartes, with the hope that the brilliant mathematicians would cooperate, in this case he failed. In fact, the two later became opponents divided by their conflicting religious views. Mersenne also expended a lot of effort promoting the work of Galileo to others in his group, even offering to translate and publish Galileo’s work in French, an offer that the Tuscan mathematician declined. He did, however, publish an unpublished text by Galileo on mechanics, Les Mechaniques de Galilée.

Although not the author of a big theory or big idea, or the instigators of a big event, Mersenne actually contributed with his activities at least as much, if not more, to the development of science in the seventeenth century as any of the more famous big names. If we really want to understand how science develops then we need to pay more attention to figures like Mersenne and turn down the volume on the big names. 


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

A seventeenth century Jesuit, who constructed his own monument and designed the first(?) ‘auto-mobile’.

One of the world’s great tourist attractions is the Imperial Observatory in Beijing.

Source: Top 12 Best Places to go visiting Beijing

The man, who rebuilt it in its current impressive form was the seventeenth century Jesuit mathematician, astronomer, and engineer Ferdinand Verbiest (1623–1688).

Ferdinand Verbiest artist unknown Source: Wikimedia Commons

I have no idea how many Jesuits took part in the Chines mission in the seventeenth century[1]. A mission that is historically important because of the amount of cultural, scientific, and technological information that flowed between Europe and China in both directions. But Jean-Baptiste Du Halde’s print of the Jesuit Mission to China only shows the three most important missionaries, Matteo Ricci Johann Adam Schall von Bell and Ferdinand Verbiest.

Jesuit Mission to China, left to right  Matteo Ricci, Johann Adam Schall von Bell, Ferdinand Verbiest Source: Wikimedia

I have already written blog posts about Ricci and Schall von Bell and here, I complete the trilogy with a sketch of the life story of Ferdinand Verbiest and how, as the title states, he came to build his own monument in the form of one of the most splendid, surviving, seventeenth-century observatories. 

Ferdinand Verbiest was born 9 October 1623 in Pittem, a village about 25 km south of Bruges in the Spanish Netherlands, the fourth of seven children of the bailiff and tax collector, Judocus Verbiest and his wife Ann van Hecke. Initially educated in the village school, in 1635 was sent to school in Bruges. In 1636 he moved onto the Jesuit College in Kortrijk. In 1641 he matriculated in Lily College of the University of Leuven, the liberal arts faculty of the university. He entered the Society of Jesus 2 September 1641 and transferred to Mechelen for the next two years. In 1643 he returned to the University of Leuven for two years, where he had the luck to study mathematics under Andrea Tacquet (1612–1660) an excellent Jesuit mathematics pedagogue. 


In 1645, Verbiest became a mathematics teacher at the Jesuit College in Kortrijk, In the same year he applied to be sent to the Americas as a missionary, but his request was turned down.

One time Jesuit College now the Church of Saint Michael Kortrijk Source: Wikimedia Commons

In 1647 his third request was granted, and he was assigned to go to Mexico. However, in Spain the authorities refused him passage and he went instead to Brussels where he taught Greek and Latin from 1648 to 1652. He was now sent to the Gregorian University in Rome where he studied under Athanasius Kircher (1602–1680) and Gaspar Schott (1608–1666). In 1653, he was granted permission to become a missionary in the New Kingdom of Granada (now Columbia) but was first sent to Seville to complete his theological studies, which he did in 1655. Once again, the Spanish authorities refused him passage to the Americas, so he decided to go to China instead.

Whilst waiting for a passage to China he continued his studies of mathematics in Genoa. In 1656 he travelled to Lisbon; however, his plans were once again foiled when pirates hijacked the ship, he was due to sail on, whilst waiting for a new ship he taught mathematics at the Jesuit College in Coimbra. In 1657, he finally sailed from Lisbon eastwards with 37 missionaries of whom 17 were heading for China under the leadership of Martino Martini (1614–1661), a historian and cartographer of China, who provided the atlas of China for Joan Blaeu’s Atlas Maior, his Novus Atlas Sinensis.

Martino Martini Source: Wikimedia Commons
Frontpage of Novus Atlas sinensis, by Martino Martini, Amsterdam, 1655. Source: Wikimedia Commons

They arrived in Goa 30 January 1658 and sailed to Macao, which they reached 17 June. In the spring of 1659, now 37 years old, he finally entered China.

Verbiest was initially assigned to be a preacher in the Shaanxi province but in 1660 Johann Adam Schall von Bell (1591–1666), who was President of the Imperial Astronomical Institute and personal adviser to the Emperor Shunzhi (1638–1661), called him to Beijing to become his personal assistant. However, in 1664, following Shunzhi’s death in 1661, Schall von Bell fell foul of his political opponents at court and both he and Verbiest were thrown into jail. Because Schall von Bell had suffered a stroke, Verbiest functioned as his representative during the subsequent trial. Initially sentenced to death, they were pardoned and rehabilitated by the new young Kangxi Emperor Xuanye (1654–1722), Schall von Bell dying in 1666.  

Johann Adam Schall von Bell artist unknown Source: Wikimedia Commons

Yang Guangxian (1597–1669), Schall von Bell’s Chinese rival, took over the Directorship of the Imperial Observatory and the Presidency of the Imperial Astronomical Institute and although now free Verbiest had little influence at the court. However, he was able to demonstrate that Yang Guangxian’s calendar contained serious errors. Constructing an astronomical calendar, which was used for astrological and ritual purposes, was the principal function of the Imperial Astronomical Institute, so this was a serious problem. A contest was set up between Verbiest and Yang Guangxian to test their astronomical acumen, which Verbiest won with ease. Verbiest was appointed to replace Yang Guangxian in both of his positions and also became a personal advisor to the still young emperor.

Kangxi Emperor Xuanye (1654–1722) unknown artist Source: Wikimedia Commons

Verbiest tutored the Kangxi Emperor in geometry and a skilled linguist (he spoke Manchu, Latin, German, Dutch, Spanish, Italian, and Tartar) he translated the first six books of the Element of Euclid in Manchu for the Emperor. Matteo Ricci (1552–1610) together with Xu Guangqi (1562–1633) had translated them into Classical Chinese, the literal language of the educated elite, in 1607.

Matteo Ricci and Xu Guangqui (from Athanasius Kircher, China Illustrata, 1670). Source: Wikimedia Commons

Verbiest, like Schall von Bell before him, used his skills as an engineer to cast cannons for the imperial army,

A cannon made with technical guidance by Ferdinand Verbiest(Nan Huairen), in Hakozaki Shrine, Higashi Ward, Fukuoka City, Fukuoka, Japan. Source: Wikimedia Commons

but it was for the Imperial Observatory that he left his greatest mark as an engineer, when in 1673 he received the commission to rebuild it. 

Imperial Observatory Beijing Source: Wikimedia Commons

The Beijing Imperial Observatory was originally constructed in 1442 during the Ming dynasty. It was substantially reorganised by the Jesuits in 1644 but underwent its biggest restoration at the hands of Verbiest.

The emperor requested the priest to construct instruments like those of Europe, and in May, 1674, Verbiest was able to present him with six, made under his direction: a quadrant, six feet in radius; an azimuth compass, six feet in diameter; a sextant, eight feet in radius; a celestial globe, six feet in diameter; and two armillary spheres, zodiacal and equinoctial, each six feet in diameter. These large instruments, all of brass and with decorations which made them notable works of art, were, despite their weight, very easy to manipulate, and a credit to Verbiest’s mechanical skill as well as to his knowledge of astronomy and mathematics. They are still in a perfect state of preservation … Joseph Brucker, Ferdinand Verbiest, Catholic Encyclopedia (1913)

Childe, Thomas: Sternwarte, Peking. Observatory, Peking, c.1875. Terrace view. Source: Wikimedia Commons

Many secondary sources attribute the instrument designs to Verbiest

L0020841 Illustrations of astronomical instruments, Beijing, China Credit: Wellcome Library, London. via Wikimedia Commons

but they are, in fact, basically copies of the instruments that Tycho Brahe designed for his observatory on the island of Hven.

Tycho Brahe’s astronomical instruments from his Astronomiae instauratae progymnasmata 1572 Source:

The Jesuits were supporters of the Tychonic helio-geocentric model of the cosmos in the seventeenth century. Verbiest recreated Hven in Beijing.  

Ricci had already realised the utility of geography and cartography in gaining the interest and trust of the Chinese and using woodblocks had printed a world map with China in the centre, Kunyu Wanguo Quantu, at the request of the Wanli Emperor, Zhu Yijun, in 1602. He was assisted by the Mandarin Zhong Wentao and the technical translator Li Zhizao. It was the first western style Chinese map. 

Kunyu Wanguo Quantu Left panel Source Wikimedia Commons
Kunyu Wanguo Quantu Right panel Source: Wikimedia Commons

In 1674, Verbiest once again followed Ricci’s example and printed, using woodblocks, his own world map the Kunya Quantu, this time in the form of two hemispheres, with the Americas in the right-hand hemisphere and Asia, Africa, and Europe in the left-hand one, once again with China roughly at the centre where the two meet.

Kunyu Quantu Source: Wikimedia Commons

It was part of a larger geographical work the Kunyu tushuo as Joseph Brucker describes it in his Catholic Encyclopedia article (1907):

the map was part of a larger geographical work called ‘Kunyu tushuo’ (Illustrated Discussion of the Geography of the World), which included information on different lands as well as the physical map itself. Cartouches provide information on the size, climate, land-forms, customs and history of various parts of the world and details of natural phenomena such as eclipses and earthquakes.  Columbus’ discovery of America is also discussed.  Images of ships, real and imaginary animals and sea creatures pepper both hemispheres, creating a visually stunning as well as historically important object.

Due to his success at gaining access to the imperial court and the emperor, in 1677, Verbiest was appointed vice principle that is head of the Jesuit missions to China, a position that he held until his death.

Perhaps the most fascinating of all of Verbiest creations was his ‘auto-mobile’, which he built for Kangxi sometime tin the 1670s.

The steam ‘car’ designed by Verbiest in 1672 – from an 18th-century print Source: Wikimedia Commons

L. H. Weeks in his Automobile Biographies. An Account of the Lives and the Work of Those Who Have Been Identified with the Invention and Development of Self-Propelled Vehicles on the Common Roads (The Monograph Press, NY, 1904) describes it thus:

The Verbiest model was for a four-wheeled carriage, on which an aeolipile was mounted with a pan of burning coals beneath it. A jet of steam from the aeolipile impinged upon the vanes of a wheel on a vertical axle, the lower end of the spindle being geared to the front axle. An additional wheel, larger than the supporting wheels, was mounted on an adjustable arm in a manner to adapt the vehicle to moving in a circular path. Another orifice in the aeolipile was fitted with a reed, so that the steam going through it imitated the song of a bird.

The aeolipile was steam driving toy described in the Pneumatica of Hero of Alexandria and the De architectura of Vitruvius, both of which enjoyed great popularity in the sixteenth and seventeenth centuries in Europe. 

A modern replica of Hero’s aeolipile. Source: Wikimedia Commons

Having suffered a fall while out horse riding a year before, Verbiest died on 28 January 1688 and was buried with great ceremony in the same graveyard as Ricci and Schall von Bell. A man of great learning and talent he forged, for a time, a strong link between Europe and China. For example, Verbiest correspondence and publications were the source of much of Leibniz’s fascination with China. He was succeeded in his various positions by the Belgian Jesuits, mathematician and astronomer Antoine Thomas (1644–1709), whom he had called to Beijing to be his assistant in old age as Schall von Bell had called him three decades earlier. 

[1] According to research by David E. Mungello from 1552 (i.e., the death of St. Francis Xavier) to 1800, a total of 920 Jesuits participated in the China mission, of whom 314 were Portuguese, and 130 were French. Source: Wikipedia


Filed under Early Scientific Publishing, History of Astronomy, History of Mathematics, History of science

They also serve…

In 1610, Galileo published his Sidereus Nuncius, the first publication to make known the new astronomical discoveries made with the recently invented telescope.

Source: Wikimedia Commons

Although, one should also emphasise that although Galileo was the first to publish, he was not the first to use the telescope as an astronomical instrument, and during that early phase of telescopic astronomy, roughly 1609-1613, several others independently made the same discoveries. There was, as to be expected, a lot of scepticism within the astronomical community concerning the claimed discoveries. The telescopes available at the time were generally of miserable quality and Galileo’s discoveries proved difficult to replicate. It was the Jesuit mathematicians and astronomers in the mathematics department at the Collegio Romano, who would, after initial difficulties, provide the scientific confirmation that Galileo desperately needed. The man, who led the endeavours to confirm or refute Galileo’s claims was the acting head of the mathematics faculty Christoph Grienberger (the professor, Christoph Clavius, was old and infirm). Grienberger is one of those historical figures, who fades into the background because they made no major discoveries or wrote no important books, but he deserves to be better known, and so this brief sketch of the man and his contributions.

As is all too oft the case with Jesuit scholars in the Early Modern Period, we know almost nothing about Grienberger before he joined the Jesuit Order. There are no know portraits of him. The problems start with his name variously given as Bamberga, Bamberger, Banbergiera, Gamberger, Ghambergier, Granberger, Panberger and a total of nineteen variations, history has settled on Grienberger. He was born 2 July 1561 in Hall a small town in the Tyrol in the west of Austria. That’s all we know till he entered the Jesuit Order in 1580. He studied rhetoric and philosophy in Prague from 1583 to 1584. From 1587 he was a mathematics teacher at the Jesuit university in Olmütz. He began his theology studies, standard for a Jesuit, in Vienna in 1589, also teaching mathematics. His earliest surviving letter to Christoph Clavius, who he had never met but who he describes as his teacher, he had studied the mathematical sciences using Clavius’ books, is dated from 1590. In 1591 he moved to the Collegio Romano, where he became Clavius’ deputy. 

In 1595, Clavius went to Naples, the purpose of his journey is not clear, but he was away from Rome for somewhat more than a year. During his absence Grienberger took over direction of the mathematics department at the Collegio Romana. From the correspondence between the two mathematicians, during this period, it becomes very obvious that Grienberger does not enjoy being in the limelight. He complains to Clavius about having to give a commencement speech and also about having to give private tuition to the sons of aristocrats. Upon Clavius’ return he fades once more into the background, only emerging again with the commotion caused by the publication of Galileo’s Sidereus Nuncius.

Rumours of Galileo’s discoveries were already making the rounds before publication and, in fact on the day the Sidereus Nuncius appeared, the wealthy German, Humanist Markus Welser (1558–1614) from Augsburg wrote to Clavius asking him his opinion on Galileo’s claims.

Markus Welser artist unknown Source: Wikimedia Commons

We know from letters that the Jesuit mathematicians in the Collegio Romano already had a simple telescope and were making astronomical observations before the publication of the Sidereus Nuncius. They immediately took up the challenge of confirming or refuting Galileo’s discoveries. However, their telescope was not powerful enough to detect the four newly discovered moons of Jupiter. Grienberger was away in Sicily attending to problems at the Jesuit college there, so it was left to Giovanni Paolo Lembo (c. 1570–1618) to try and construct a telescope good enough to complete the task. We know that Lembo was skilled in this direction because between 1615 and 1617 he taught lens grinding and telescope construction to the Jesuits being trained as missionaries to East Asia at the University of Coimbra. 

Lembo’s initial attempts to construct a suitable instrument failed and it was only after Grienberger returned from Sicily that the two of them were able to make progress. At this point Galileo was corresponding with Clavius and urging the Jesuit astronomer on provide the confirmation of his discoveries that he so desperately needed, the general scepsis was very high, but he was not prepared to divulge any details on how to construct good quality telescopes. Eventually, Grienberger and Lembo succeeded in constructing a telescope with which they could observe the moons of Jupiter but only under very good observational conditions. They first observed three of the moons on 14 November 1610 and all four on 16 November. 

Clavius wrote to the merchant and mathematician Antonio Santini (1577–1662) in Venice, who had been to first to confirm the existence of the Jupiter moons in 1610, with a telescope that he constructed himself, detailing observation from 22, 23, 26, and 27 November but stating that they were still not certain as to the nature of the moons. Santini relayed this information to Galileo. On 17 December, Clavius wrote to Galileo:

…and so we have seen [the Medici Stars] here in Rome many times. At the end of the letter I will put some observations, from which it follows very clearly that they are not fixed but wandering stars, because they change position with respect to each other and Jupiter.

Much of what we know about the efforts of the Jesuit astronomers under the leadership of Grienberger to build an adequate telescope to confirm Galileo’s discoveries come from a letter that Grienberger wrote to Galileo in February 1611. One interesting aspect of Grienberger’s letter is that the Jesuit astronomers had also been observing Venus and there is good evidence that they discovered the phases of Venus independently at least contemporaneously if not earlier than Galileo. This was proof that Venus, and by analogy probably also Mercury, orbit the Sun and not the Earth. This was the death nell for a pure Ptolemaic geocentric system and the acceptance at a minimum of a Capellan system where the two inner planets orbit the Sun, which orbits the Earth, if not a full blown Tychonic system or even a heliocentric one. This was in 1611 troubling for the conservative leadership of the Jesuit Order, but would eventually lead to them adopting a Tychonic system at the beginning of the 1620s. 

Clavius died 6 February 1612 and Grienberger became his official successor as the professor for mathematics at the Collegio Romano, a position he retained until 1633, when he was succeeded in turn by Athanasius Kircher (1602–1680). The was a series of Rules of Modesty in Ignatius of Loyola’s rules for the Jesuit Order and individual Jesuits were expected to self-abnegate. The most extreme aspect of this was that many scientific works were published anonymously as a product of the Order and not the individual. Different Jesuit scholars reacted differently to this principle. On the one hand, Christoph Scheiner (1573–1650), Galileo’s rival in the sunspot dispute and author of the Rosa Ursina sive Sol(1626–1630) presented himself as a great astronomer, which did not endear him to his fellow Jesuits.

Christoph Scheiner artist unknown Source: Wikimedia Commons

On the other hand, Grienberger put his name on almost none of his own work preferring it to remain anonymous. There is only a star catalogue and a set of trigonometrical tables that bear his name.

However, as head of the mathematics department at the Collegio Romano he was responsible for controlling and editing all of the publications in the mathematical disciplines that went out from the Jesuit Order and it is know that he made substantial improvements to the works that he edited both in the theoretical parts and in the design of instruments. A good example is the heliotropic telescope, which enables the observer to track the movement of the Sun whilst observing sunspots, illustrated in Scheiner’s Rosa Ursina.

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

This instrument is known to have been designed and constructed by Grienberger, who, however, explicitly declined Scheiner’s offer to add a text under his own name describing its operation. Grienberger also devised a system of gearing theoretical capable of lifting the Earth

Reconstruction of Grienberger’S Earth lifting gearing

Grienberger, admired Galileo and took his side, if only in the background, in Galileo’s dispute with the Aristotelians over floating bodies. He was, however, disappointed by Galileo’s unprovoked and vicious attacks on the Jesuit astronomer Orazio Grassi on the nature of comets and explicitly said that it had cost Galileo the support of the Jesuits in his later troubles. He also clearly stated that if Galileo had been content to propose heliocentricity as a hypothesis, its actual scientific status at the time, he could have avoided his confrontation with the Church.

Élie Diodati (1576–1661) the Calvinist, Genevan lawyer and friend of Galileo, who played a central role in the publication of the Discorsi, quoted Grienberger in a letter to Galileo from 25 July 1634, as having said, “If Galileo had recognised the need to maintain the favour of the Fathers of this College, then he would live gloriously in the world, and none of his misfortune would have occurred, and he could have written about any subject, as he thought fit, I say even about the movement of the Earth…”

Several popular secondary sources claim the Grienberger supported the Copernican system. However, there is only hearsay evidence for this claim and not actual proof. He might have but we will never know. 

Grienberger made no major discoveries and propagated no influential new theories, which would launch him into the forefront of the big names, big events style of the history of science. However, he played a pivotal role in the very necessary confirmation of Galileo’s telescopic discoveries. He also successfully helmed the mathematical department of the Collegio Romano for twenty years, which produced many excellent mathematicians and astronomers, who in turn went out to all corners of the world to teach others their disciplines. By the time Athanasius Kircher inherited Grienberger’s post there was a world-wide network of Jesuit astronomers, communicating data on important celestial events. One such was Johann Adam Schall von Bell (1591–1666), who studied under Grienberger and went on to lead the Jesuit mission in China.

Johann Adam Schall von Bell Source: Wikimedia Commons

Science is a collective endeavour and figures such as Grienberger, who serve inconspicuously in the background are as important to the progress of that endeavour as the shrill public figures, such as Galileo, hogging the limelight in the foreground. 


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

Renaissance Science – XIII

As already explained in the fourth episode of this series, the Humanist Renaissance was characterised by a reference for classical literature, mostly Roman, recovered from original Latin manuscripts and not filtered and distorted through repeated translations on their way from Latin into Arabic and back into Latin. It was also a movement that praised a return to classical Latin, away from the, as they saw it, barbaric medieval Latin. In the fifth episode we also saw that, what I am calling, Renaissance science was characterised by a break down of the division that had existed between theoretical book knowledge as taught on the medieval universities and the empirical, practical knowledge of the artisans. As also pointed there this was not so much a breaking down of boundaries or a crossover between the two fields of knowledge as a meld between the two types of knowledge that would over the next two and a half centuries lead to the modern concept of knowledge or science.

One rediscovered classical Latin text that very much filled the first criterium, which at the same time played a major role in the second was De architectura libri decem (Ten Books on Architecture) by the Roman architect and civil and military engineer Marcus Vitruvius Pollio (c.80-70–died after 15 BCE), who is usually referred to simply as Vitruvius and there are doubts about the other two names that are ascribed to him. 

From the start we run into problems about the standard story that the manuscript was rediscovered by the Tuscan, humanist scholar Poggio Bracciolini (1380–1459) in the library of Saint Gall Abbey in 1416, as related by Leon Battista Alberti (1404–1472) in his own architecture treatise De re aedificatoria (1452), which was modelled on Vitruvius’ tome. In reality, De architectura had never been lost during the Middle Ages; there are about ninety surviving medieval manuscripts of the book.

Manuscript of Vitruvius; parchment dating from about 1390 Source: Wikimedia Commons

The oldest was made during the Carolingian Renaissance in the early nineth century. Alcuin of York was consulted on the technical terms in the text. During the Middle Ages many leading scholars including Hermann of Reichenau (1013–1054), a central figure of the Ottonian Renaissance, and both Albertus Magnus (c. 1200– 1280) and Thomas Aquinas (1225–1274), who laid the foundations of medieval Aristotelian philosophy, read the text, and commented on it. 

However, although well-known it had little impact on architecture in the medieval period. The great medieval cathedrals and castle were built by master masons, whose knowledge was practical artisanal knowledge passed on by word of mouth from master to apprentice. This changed with Poggio’ rediscovery of Vitruvius’ work and the concept of the theoretical and practical architect began to emerge.

Before we turn to the impact of De architectura in the Renaissance we first need to look at the book and its author. Very little is known about Vitruvius, as already stated above, the other names attributed to him are based on speculation, most of what we do know is pieced together from the book itself. Vitruvius was a military engineer under Gaius Julius Caesar (100–44 BCE) and apparently received a pension from Octavian (63 BCE–14 CE), the later Caesar Augustus, to whom the book is dedicated. The book was written around twenty BCE. Vitruvius wrote it because he believed in making knowledge public and available to all, unlike those artisans, who kept their knowledge secret.

The ten books are organised as follows:

  1. Town planning, architecture or civil engineering in general and the qualification required by an architect or civil engineer
  2. Building materials
  3. Temples and the orders of architecture
  4. As book 3
  5. Civil buildings
  6. Domestic buildings
  7. Pavements and decorative plasterwork
  8. Water supplies and aqueducts
  9. The scientific side of architecture – geometry, measurement, astronomy, sundials
  10. Machines, use and construction – siege engines, water mills, drainage machines, technology, hoisting, pneumatics

In terms of its reception and influence during the Renaissance the most important aspect is Vitruvius’ insistence that architecture requires both ratiocinatio and fabrica, that is reasoning or theory, and practice or construction. This Vitruvian philosophy of architecture took architecture out of the exclusive control of the master mason and into the hands of the theoretical scholars in union with the artisans. This move was also motivated by the humanist drive to study archaeologically the Roman remains in Rome the Eternal City. Vitruvius provided a guide to understanding the Roman architecture, which would become the model for the construction of new buildings. 

But for it to become influential Vitruvius’s text first had to become widely available. The first printed Latin edition was edited by the humanist scholar Fra. Giovanni Sulpizio da Veroli (fl. c. 1470–1490) and published in 1486 with a second edition in 1495 or 1496.


The first printed edition had no illustrations. Fra. Giovanni Giocondo da Verona (c. 1433–1515) produced the first edition with woodcut illustrations, published in Venice in 1511. A second improved edition was published in Florence in 1521. 


In order for De architectura to reach artisans it needed to be translated into the vernacular, as most of them couldn’t read Latin. This process began in Italy and during the sixteenth century spread throughout Europe. The process started already before De architectura appeared in print. As mentioned above Alberti’s De re aedificatoria (On the Art of Buildings), not a translation of De architectura but a book strongly modelled on it appeared in Latin in print in 1452.

Source: Wikimedia Commons

The first Italian edition appeared in 1486 A second Italian edition, by the humanist mathematician Cosimo Bartoli (1503-1572), which became the standard edition, appeared in 1550. Alberti was very prominent in Renaissance culture and very widely read. His influence can be measured by the fact that a collective bilingual, English/Italian, edition of his works on architecture, painting and sculpture was published as late as 1726. 

The first Italian edition of De architectura with new illustration and added commentary by Cesare Cesariano (1475-1543) was published at Como in 1521.

1521 Italian edition title page Source
1521 Italian edition

A plagiarised version was published in Venice in 1524. The first French edition, translated by Jean Martin (died 1553), which is said to contain many errors, was published in Paris in 1547.


The first German edition was translated by Walther Hermann Ryff (c. 1500–1548). As far as can be determined, it appears the Ryff was an apothecary but work mostly as what today would probably be described as a hack. He published as editor, translator, adapter, and compiler a large number of books, around 40, over a wide range of topics, although the majority were in some sense medical, and was seemingly very successful. He was often accused of plagiarism. The physician and botanist, Leonhart Fuchs (1501–1566) described him as an “extremely brazen, careless, fraudulent author.” Apart from his medical works, Ryff obviously had a strong interest in architecture. He edited and published a Latin edition of De architectura in Strasbourg in 1543. This was followed by a commentary on De architectura in German, Der furnembsten, notwendigsten, der gantzen Architectur angehörigen Mathematischen vnd Mechanischen künst, eygentlicher bericht, vnd vast klare, verstendliche vnterrichtung, zu rechtem verstandt der lehr Vitruuij, in drey furneme Bücher abgetheilet (The most distinguished, necessary, mathematical and mechanical arts belonging to the entire architecture, actual report and clear, understandable instruction of the teachings of Vitruvius shared in three distinguished books), published by Johannes Petreius, the leading European scientific publisher of the period, in Nürnberg in 1547. For obvious reasons this is usually simply referred to as Architektur. This was obviously a product of the German translation of De architectura, which Petreius had commissioned Ryff to produce and, which he published in Nürnberg in 1548 under the title, Vitruvius Teutsch. Nemlichen des aller namhafftigisten vñ hocherfahrnesten römischen Architecti vnd kunstreichen Werck zehn Bücher von der Architectur und künstlichem Bawen… (Vitruvius in German…).


We return now to Italy and the story of the stone mason, Andrea di Pietro della Gondola, born in Padua in 1508. Having served his apprenticeship, he worked as a stone mason until he was thirty years old. In 1538–39, he was employed to rebuild the villa of the humanist poet and scholar, Gian Giorgio Trissino (1478–1550) to rebuild his villa in Cricoli.

Gian Giorgio Trissino, portrayed in 1510 by Vincenzo Catena Source: Wikimedia Commons
Villa Trissino Source: Wikimedia Commons

Trissino ran a small private learned academy for young gentlemen in his renovated villa and apparently, having taken a shine to the young stone mason invited him to become a member. Andrea accepted the offer and Trissino renamed him Palladio.

Portrait of Palladio by Alessandro Maganza Source: Wikimedia Commons

The two became friends and colleagues, and Trissino, who was deeply interested in classical architecture and Vitruvius took the newly christened Palladio with him on trips to Rome to study the Roman ruins. Palladio became an architect in 1540 and became a specialist for designing and building neo-classical, Palladian, villas. 

Villa Barbaro begun 1557 Source: Wikimedia Commons

Trissino died in 1550 but Palladio acquired a new patron, Daniele Barbaro (1514–1570), a member of one of the most prominent and influential aristocratical families of Venice.

Daniele Barbaro by Paolo Veronese (the book in the painting is Barbaro’s translation of De architectura)

Daniele Barbaro studied philosophy, mathematics, and optics at the University of Padua. He was a diplomat and architect, who like Trissino, before him, accompanied Palladio on expeditions to study Roman architecture. In 1556, Barbaro published a new Italian translation of De architectura with an extended commentary, Dieci libri dell’architettura di M. Vitruvio.

I dieci libri dell’architettura di M. Vitruvio tradutti et commentati da monsignor Barbaro eletto patriarca d’aquileggia 1556 Images by Palladio Source

In 1567, he, simultaneously published, a revised Italian and a Latin edition entitled M. Vitruvii de architectura. The illustrations for Barbaro’s editions were provided by Palladio. Barbaro provided the best, to date, explanations of much of the technical terminology in De architectura, also acknowledging Palladio’s theoretical contributions to the work.

Palladio had become one of the most important and influential architects in the whole of Europe, designing many villas, palaces, and churches. He also became an influential author publishing L’Antichida di Roma (The Antiquities of Rome) in 1554,


and I quattro libri dell’architettura (The Four Books of Architecture) in 1570, which was heavily influenced by Vitruvius. His books were translated into many different languages and went through many editions right down into the eighteenth and nineteenth centuries. His work inspired leading architects in France and Germany.

Title page from 1642 edition Source: Wikimedia Commons

Up till now we have said nothing about England, which as usual lagged behind the continent in things mathematical, although in the second half of the sixteenth century both Leonard Digges and John Dee, of the so-called English school of mathematics, counted architecture under the mathematical disciplines. In 1563 John Shute (died 1563) included Vitruvian elements in his The First and Chief Grounds of Architecture.

John Shute The First and Chief Grounds of Architecture.

Inigo Jones (1573–1652) was born into a Welsh speaking family in Smithfield, London. There is minimal evidence that he was an apprentice joiner but at some point, before 1603 he acquired a rich patron, who impressed by his sketches, sent him to Italy to study drawing in Italy.

Inigo Jones by Anthony van Dyck

In a second visit to Italy in 1606 he came under the influence of Sir Henry Wotton (1568–1639) the English ambassador to Venice.

Henry Wotton artist unknown Source: Wikimedia Commons

Wotton was interested in astronomy, and it was he, who sent two copies of Galileo’s Sidereus Nuncius (1610) to London on the day it was published. Wotton convinced Jones to learn Italian and introduced him to Palladio’s I quattro libri dell’architettura. Jones’ copy of the book has marginalia that references Wotton. In 1624, Wotton published The Elements of Architecture a loose translation of De architectura into English. The first proper translation appeared only in 1771. 

19th century reprint Source

Inigo Jones introduced the Vitruvian–Palladian architecture into England and became the most influential architect in the country, becoming Surveyor of the King’s Works.

The Queen’s House in Greenwich designed and built by Inigo Jones Source: Wikimedia Commons

His career was ended with the outbreak of the English Civil War in 1642. England’s most famous architect Christopher Wren (1632–1723), a mathematician and astronomer turned architect stood in a line with Vitruvius, Palladio, and Jones. It is very clear that the humanist rediscovery and promotion of De architectura had a massive influence on the development of architecture in Europe in the sixteenth and seventeenth centuries, in the process dissolving the boundaries between the theoretical intellectuals and the practical artisans. 


Filed under History of Mathematics, History of Technology, Renaissance Science