Category Archives: Renaissance Science

The emergence of modern astronomy – a complex mosaic: Part XXXIII

In the previous episode of this series we looked into the academic literature that spread knowledge of the heliocentric system during the seventeenth century. However, there was another genre of literature during the century that was also partially dedicated to introducing and explaining the heliocentric system, fiction and popular literature and that is what we are going to look at now.

It should come as no surprise that the earliest author to produce a fictional account of the heliocentric system was Johannes Kepler with his posthumously published proto-science-fiction novel, Somnium (The Dream) (1634).

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Source: Wikimedia Commons

Kepler first wrote the core of this book as a student dissertation, written for his teacher Michael Mästlin, explaining how the movement of the Earth, in a heliocentric system, would appear to somebody observing it from the Moon. Around 1605 he added a frame story to his student dissertation the dream of the title. Kepler relates that in 1608 he was reading a book on Bohemian legends when he fell asleep and began to dream. In his dream he takes down another book from the shelf and reads the story of Duracotus, an Icelander, and his mother Fiolxhilda, who is obviously a witch, although Kepler never explicitly states that. The boy open a herb charm that his mother has made to sell to sailors and removes the herbs making the charm useless. Outraged, his mother sells him instead to the ship’s captain, who takes him to Scandinavia, where he ends up on Hven with Tycho Brahe under whom he studies astronomy for five years. Returning to Iceland he reconciles himself with his mother, who reveals to him that she has magical knowledge of astronomy. Fiolxhilda summons a daemon, who tells Duracotus how they could travel to the moon and then holds a long discourse on the moon and its inhabitants, part science, part science fiction. To go into more detail would turn this post into book, however, because of the obvious autobiographical element Kepler thought that somebody had gained access to the manuscript and this was why his mother was charged with witchcraft; he was almost certainly mistaken in this belief.

Kepler did not publish his story but put it aside. Between 1620 and 1630 Kepler added 223 extensive endnotes, which elucidate the story, explaining his sources, his motivations and the content of the story itself. Even with these explanatory additions Kepler did not publish the book, leaving it unpublished at his death in 1630. Because his death had left his wife, Susanna, and his family in financial difficulties, his son in law, Jacob Bartsch (c. 1600–1633) edited the manuscript for publication with hope of generating an income for his mother-in-law. However, he too died before he could publish the book, which was then finally brought to press by Kepler’s son Ludwig (b. 1607).

Kepler’s Somnium was the first of a series of fictional books describing journeys to the moon in the seventeenth century nearly all of which promoted a heliocentric astronomy and it is to these that we now turn.

Our first author is the Anglican clergyman and natural philosopher, John Wilkins (1614–1672).

Greenhill, John, c.1649-1676; John Wilkins (1614-1672), Warden (1648-1659)

John Wilkins portrait attributed to John Greenhill Source: Wikimedia Commons

Although he produced no real new scientific discoveries or theories Wilkins was a highly influential figure in the scientific revolution in England. He published a series of popular and speculative science books and was a founding member of and a driving force behind the Royal Society. One of Wilkins’ popular science books, Mathematical Magick (1648) is said to have had a strong influence on a young Isaac Newton but it is two of his other books that interest us here, The Discovery of a World in the Moone (1638) and A Discourse Concerning a New Planet (1640), the second being a revised and expanded version of the first.

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Cover and frontispiece. Note the heliocentric system diagram

Both books present a heliocentric astronomical system and, based on Galileo’s telescopic discoveries of the earth like nature of the moon, hypothesise an inhabited moon, as had Kepler’s Somnium, which however predated Galileo’s Sidereus Nuncius. Wilkins two books were a popular source for disseminating the heliocentric hypothesis in England.

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Five months after the publication of The Discovery of a World in the Moone another journey to the moon fantasy by an Anglican clergyman was published, Francis Goodwin’s The Man in the Moone or A Discourse of a Voyage thither (1638), under the pseudonym Domingo Gonsales.

unknown artist; Francis Godwyn (1562-1633), Bishop of Llandaff (1601), Bishop of Hereford (1617)

Francis Godwin artist unknown Source: Wikimedia Commons

Godwin (1562–1633) had died five years previously and although his book was published after Wilkins’ tome, it is thought to have been written in the 1620s and it is known to have influenced Wilkins’ book.

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Source: Houghton Library via Wikimedia Commons

The ‘author’, Gonsales, a Spaniard on the run after killing a man in a duel, invents a flying machine powered by gansa, a species of wild swans, which after a series of adventures flies him to the moon, a twelve day journey. Here he discovers a utopian Christian society. After six months he returns to earth landing in China, where he has some more adventures. For our purposes what is important here is that like Wilkins, Godwin is a Copernican and although he only mentions Copernicus by name the influence of Kepler, Gilbert and Galileo is clearly discernable in his science fantasy.

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Frontispiece and title page of the second edition (1657), now with the pseudonym replaced by “F.G. B. of H.” (“Francis Godwin, Bishop of Hereford”) Source: Wikimedia Commons

The books of Wilkins and Godwin were both best sellers and were translated into various other European languages including French, where they influenced another book in the genre, Cyrano de Bergerac’s L’Autre monde ou les états et empires de la Lune (The Other World: Comical History of the States and Empires of the Moon 1657), and his Les États et Empires du Soleil (The States and Empires of the Sun, 1662), both published posthumously.

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Cyrano de Bergerac artist unknown Source: Wikimedia Commons

L’Autre monde is a satire on Godwin’s book and Cyrano’s hero, who is also called Cyrano, makes various failed attempts to reach the moon, including trying to rise up to the moon levitated by bottles of evaporating dew before he finally gets there.

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Cyrano uses bottles of dew to float upwards. Illustration from the second volume of an edition of Cyrano de Bergerac’s complete works printed in Amsterdam in 1708 Source: Wikimedia Commons

When he does arrive on the moon one of the people he meets is Gonsales, with whom he has a religious debate. It might seem that Cyrano de Bergerac (1619–1655) as a literary author was just riffing off the success of Wilkins’ and Godwin’s works but he was a pupil of Pierre Gassendi (1592–1655) and so was well informed about the ongoing cosmology and astronomy debate.

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Godwin’s The Man in the Moone and the English translation of Cyrano’s L’Autre monde inspired two later stage productions on the theme Aphra Behn’s (1640–1689) farce The Emperor of the Moon 1687, her second most successful play, and Elkanah Settle’s (1648–1724) opera The World in the Moon (1697).

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Aphra Behn by the Anglo-Dutch artist Sir Peter Lely, Courtesy of the Yale Center for British Art, Yale University via Wikimedia Commons

All of the texts that we have looked at so far contain a common theme that emerged strongly during the seventeenth century, the possibility of life on other worlds, in this case the moon. Our final work, in this case not a fictional but a factual one, continues this theme, Bernard Le Bovier de Fontenelle’s popular presentation of the heliocentric hypothesis, Entretiens sur la pluralité des mondes (Conversations on the Plurality of Worlds, 1686). Bernard Le Bovier de Fontenelle (1657–1757) was an author and Cartesian philosopher, commentator rather than initiator, who was a member of both the Académie française and the Académie des sciences of which he was secretary for forty-two years beginning in 1697; in this function he wrote Histoire du renouvellement de l’Académie des Sciences (Paris, 3 vols., 1708, 1717, 1722).

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Bernard Le Bovier de Fontenelle artist unknown Source: Wikimedia Commons

His Entretiens sur la pluralité des mondes was an early example of a popular science book written in French not Latin.

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In the preface, Fontenelle addresses female readers and suggests that the offered explanation should be easily understood even by those without scientific knowledge. The book is presented as a dialogue between a philosopher and a marquise and elucidates the heliocentric system with a discussion of the possibility of extra-terrestrial life. The book is interesting in that Fontenelle explains that there is now only one system to consider because the Tychonic system was now considered to be too complex in comparison with the heliocentric system. This is one of the few real applications of Ockham’s razor in the history of science and comes long before there was any empirical proof for the heliocentric system. There was an English translation by John Glanville (c. 1664–1735) in 1687 and another by Aphra Behn A Discovery of New Worlds in 1688.

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A Gentleman of the Inner Temple is John Glanville

This all too brief survey of the fictional and popular literature published in the seventeenth century demonstrates that the discussion on the cosmological/astronomical system had escaped the narrow confines of academia and entered the public forum.

 

 

 

 

 

 

 

 

 

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How Renaissance Nürnberg became the Scientific Instrument Capital of Europe

This is a writen version of the lecture that I was due to hold at the Science and the City conference in London on 7 April 2020. The conference has for obvious reasons been cancelled and will now take place on the Internet. You can view the revised conference program here.

The title of my piece is, of course, somewhat hyperbolic, as far as I know nobody has ever done a statistical analysis of the manufacture of and trade in scientific instruments in the sixteenth century. However, it is certain that in the period 1450-1550 Nürnberg was one of the leading European centres both for the manufacture of and the trade in scientific instruments. Instruments made in Nürnberg in this period can be found in every major collection of historical instruments, ranging from luxury items, usually made for rich patrons, like the column sundial by Christian Heyden (1526–1576) from Hessen-Kassel

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Column Sundial by Christian Heyden Source: Museumslandschaft Hessen-Kassel

to cheap everyday instruments like this rare (rare because they seldom survive) paper astrolabe by Georg Hartman (1489–1564) from the MHS in Oxford.

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Paper and Wood Astrolabe Hartmann Source: MHS Oxford

I shall be looking at the reasons why and how Nürnberg became such a major centre for scientific instruments around 1500, which surprisingly have very little to do with science and a lot to do with geography, politics and economics.

Like many medieval settlements Nürnberg began simply as a fortification of a prominent rock outcrop overlooking an important crossroads. The first historical mention of that fortification is 1050 CE and there is circumstantial evidence that it was not more than twenty or thirty years old. It seems to have been built in order to set something against the growing power of the Prince Bishopric of Bamberg to the north. As is normal a settlement developed on the downhill slopes from the fortification of people supplying services to it.

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A fairly accurate depiction of Nürnberg from the Nuremberg Chronicle from 1493. The castles (by then 3) at the top with the city spreading down the hill. Large parts of the inner city still look like this today

Initially the inhabitants were under the authority of the owner of the fortification a Burggraf or castellan. With time as the settlement grew the inhabitants began to struggle for independence to govern themselves.

In 1200 the inhabitants received a town charter and in 1219 Friedrich II granted the town of Nürnberg a charter as a Free Imperial City. This meant that Nürnberg was an independent city-state, which only owed allegiance to the king or emperor. The charter also stated that because Nürnberg did not possess a navigable river or any natural resources it was granted special tax privileges and customs unions with a number of southern German town and cities. Nürnberg became a trading city. This is where the geography comes into play, remember that important crossroads. If we look at the map below, Nürnberg is the comparatively small red patch in the middle of the Holy Roman Empire at the beginning of the sixteenth century. If your draw a line from Paris to Prague, both big important medieval cities, and a second line from the border with Denmark in Northern Germany down to Venice, Nürnberg sits where the lines cross almost literally in the centre of Europe. Nürnberg also sits in the middle of what was known in the Middle Ages as the Golden Road, the road that connected Prague and Frankfurt, two important imperial cities.

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You can also very clearly see Nürnberg’s central position in Europe on Erhard Etzlaub’s  (c. 1460–c. 1531) pilgrimage map of Europe created for the Holy Year of 1500. Nürnberg, Etzlaub’s hometown, is the yellow patch in the middle. Careful, south is at the top.

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Over the following decades and centuries the merchant traders of Nürnberg systematically expanded their activities forming more and more customs unions, with the support of various German Emperors, with towns, cities and regions throughout the whole of Europe north of Italy. Nürnberg which traded extensively with the North Italian cities, bringing spices, silk and other eastern wares, up from the Italian trading cities to distribute throughout Europe, had an agreement not to trade with the Mediterranean states in exchange for the Italians not trading north of their northern border.

As Nürnberg grew and became more prosperous, so its political status and position within the German Empire changed and developed. In the beginning, in 1219, the Emperor appointed a civil servant (Schultheis), who was the legal authority in the city and its judge, especially in capital cases. The earliest mention of a town council is 1256 but it can be assumed it started forming earlier. In 1356 the Emperor, Karl IV, issued the Golden Bull at the Imperial Diet in Nürnberg. This was effectively a constitution for the Holy Roman Empire that regulated how the Emperor was to be elected and, who was to be appointed as the Seven Prince-electors, three archbishops and four secular rulers. It also stipulated that the first Imperial Diet of a newly elected Emperor was to be held in Nürnberg. This stipulation reflects Nürnberg’s status in the middle of the fourteenth century.

The event is celebrated by the mechanical clock ordered by the town council to be constructed for the Frauenkirche, on the market place in 1506 on the 150th anniversary of the Golden Bull, which at twelve noon displays the seven Prince-electors circling the Emperor.

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Mechanical clock on the Frauenkirche overlooking the market place in Nürnberg. Ordered by the city council in 1506 to celebrate the 150th anniversary of the issuing of the Golden Bull at the Imperial Diet in 1356

Over time the city council had taken more and more power from the Schultheis and in 1385 they formally bought the office, integrating it into the councils authority, for 8,000 gulden, a small fortune. In 1424 Emperor, Sigismund appointed Nürnberg the permanent residence of the Reichskleinodien (the Imperial Regalia–crown, orb, sceptre, etc.).

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The Imperial Regalia

This raised Nürnberg in the Imperial hierarchy on a level with Frankfurt, where the Emperor was elected, and Aachen, where he was crowned. In 1427, the Hohenzollern family, current holders of the Burggraf title, sold the castle, which was actually a ruin at that time having been burnt to the ground by the Bavarian army, to the town council for 120,000 gulden, a very large fortune. From this point onwards Nürnberg, in the style of Venice, called itself a republic up to 1806 when it was integrated into Bavaria.

In 1500 Nürnberg was the second biggest city in Germany, after Köln, with a population of approximately 40,000, about half of which lived inside the impressive city walls and the other half in the territory surrounding the city, which belonged to it.

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Map of the city-state of Nürnberg by Abraham Ortelius 1590. the city itself is to the left just under the middle of the map. Large parts of the forest still exists and I live on the northern edge of it, Dormitz is a neighbouring village to the one where I live.

Small in comparison to the major Italian cities of the period but even today Germany is much more decentralised with its population more evenly distributed than other European countries. It was also one of the richest cities in the whole of Europe.

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Nürnberg, Plan by Paul Pfinzing, 1594 Castles in the top left hand corner

Nürnberg’s wealth was based on two factors, trading, in 1500 at least 27 major trade routes ran through Nürnberg, which had over 90 customs unions with cities and regions throughout Europe, and secondly the manufacture of trading goods. It is now time to turn to this second branch of Nürnberg’s wealth but before doing so it is important to note that whereas in other trading centres in Europe individual traders competed with each other, Nürnberg function like a single giant corporation, with the city council as the board of directors, the merchant traders cooperating with each other on all levels for the general good of the city.

In 1363 Nürnberg had more than 1200 trades and crafts masters working in the city. About 14% worked in the food industry, bakers, butchers, etc. About 16% in the textile industry and another 27% working leather. Those working in wood or the building branch make up another 14% but the largest segment with 353 masters consisted of those working in metal, including 16 gold and silver smiths. By 1500 it is estimated that Nürnberg had between 2,000 and 3,000 trades and crafts master that is between 10 and 15 per cent of those living in the city with the metal workers still the biggest segment. The metal workers of Nürnberg produced literally anything that could be made of metal from sewing needles and nails to suits of armour. Nürnberg’s reputation as a producer rested on the quality of its metal wares, which they sold all over Europe and beyond. According to the Venetian accounts books, Nürnberg metal wares were the leading export goods to the orient. To give an idea of the scale of production at the beginning of the 16th century the knife makers and the sword blade makers (two separate crafts) had a potential production capacity of 80,000 blades a week. The Nürnberger armourers filled an order for armour for 5,000 soldiers for the Holy Roman Emperor, Karl V (1500–1558).

The Nürnberger craftsmen did not only produce goods made of metal but the merchant traders, full blood capitalists, bought into and bought up the metal ore mining industry–iron, copper, zinc, gold and silver–of Middle Europe, and beyond, (in the 16th century they even owned copper mines in Cuba) both to trade in ore and to smelt ore and trade in metal as well as to ensure adequate supplies for the home production. The council invested heavily in the industry, for example, providing funds for the research and development of the world’s first mechanical wire-pulling mill, which entered production in 1368.

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The wirepulling mills of Nürnberg by Albrecht Dürer

Wire was required in large quantities to make chainmail amongst other things. Around 1500 Nürnberg had monopolies in the production of copper ore, and in the trade with steel and iron.  Scientific instruments are also largely made of metal so the Nürnberger gold, silver and copper smiths, and toolmakers also began to manufacture them for the export trade. There was large scale production of compasses, sundials (in particular portable sundials), astronomical quadrants, horary quadrants, torquetum, and astrolabes as well as metal drawing and measuring instruments such as dividers, compasses etc.

The city corporation of Nürnberg had a couple of peculiarities in terms of its governance and the city council that exercised that governance. Firstly the city council was made up exclusively of members of the so-called Patrizier. These were 43 families, who were regarded as founding families of the city all of them were merchant traders. There was a larger body that elected the council but they only gave the nod to a list of the members of the council that was presented to them. Secondly Nürnberg had no trades and crafts guilds, the trades and crafts were controlled by the city council. There was a tight control on what could be produced and an equally tight quality control on everything produced to ensure the high quality of goods that were traded. What would have motivated the council to enter the scientific instrument market, was there a demand here to be filled?

It is difficult to establish why the Nürnberg city corporation entered the scientific instrument market before 1400 but by the middle of the 15th century they were established in that market. In 1444 the Catholic philosopher, theologian and astronomer Nicolaus Cusanus (1401–1464) bought a copper celestial globe, a torquetum and an astrolabe at the Imperial Diet in Nürnberg. These instruments are still preserved in the Cusanus museum in his birthplace, Kues on the Mosel.

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The Cusanus Museum in Kue

In fact the demand for scientific instrument rose sharply in the 15th & 16th centuries for the following reasons. In 1406 Jacopo d’Angelo produced the first Latin translation of Ptolemy’s Geographia in Florence, reintroducing mathematical cartography into Renaissance Europe. One can trace the spread of the ‘new’ cartography from Florence up through Austria and into Southern Germany during the 15th century. In the early 16th century Nürnberg was a major centre for cartography and the production of both terrestrial and celestial globes. One historian of cartography refers to a Viennese-Nürnberger school of mathematical cartography in this period. The availability of the Geographia was also one trigger of a 15th century renaissance in astronomy one sign of which was the so-called 1st Viennese School of Mathematics, Georg von Peuerbach (1423–1461) and Regiomontanus (1436–176), in the middle of the century. Regiomontanus moved to Nürnberg in 1471, following a decade wandering around Europe, to carry out his reform of astronomy, according to his own account, because Nürnberg made the best astronomical instruments and had the best communications network. The latter a product of the city’s trading activities. When in Nürnberg, Regiomontanus set up the world’s first scientific publishing house, the production of which was curtailed by his early death.

Another source for the rise in demand for instruments was the rise in interest in astrology. Dedicated chairs for mathematics, which were actually chairs for astrology, were established in the humanist universities of Northern Italy and Krakow in Poland early in the 15th century and then around 1470 in Ingolstadt. There were close connections between Nürnberg and the Universities of Ingolstadt and Vienna. A number of important early 16th century astrologers lived and worked in Nürnberg.

The second half of the 15th century saw the start of the so-called age of exploration with ships venturing out of the Iberian peninsular into the Atlantic and down the coast of Africa, a process that peaked with Columbus’ first voyage to America in 1492 and Vasco da Gama’s first voyage to India (1497–199). Martin Behaim(1459–1507), son of a Nürnberger cloth trading family and creator of the oldest surviving terrestrial globe, sat on the Portuguese board of navigation, probably, according to David Waters, to attract traders from Nürnberg to invest in the Portuguese voyages of exploration.  This massively increased the demand for navigational instruments.

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The Erdapfel–the Behaim terrestial globe Germanische National Museum

Changes in the conduct of wars and in the ownership of land led to a demand for better, more accurate maps and the more accurate determination of boundaries. Both requiring surveying and the instruments needed for surveying. In 1524 Peter Apian (1495–1552) a product of the 2nd Viennese school of mathematics published his Cosmographia in Ingolstadt, a textbook for astronomy, astrology, cartography and surveying.

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The Cosmographia went through more than 30 expanded, updated editions, but all of which, apart from the first, were edited and published by Gemma Frisius (1508–1555) in Louvain. In 1533 in the third edition Gemma Frisius added an appendix Libellus de locorum describendum ratione, the first complete description of triangulation, the central method of cartography and surveying down to the present, which, of course in dependent on scientific instruments.

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In 1533 Apian’s Instrumentum Primi Mobilis 

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was published in Nürnberg by Johannes Petreius (c. 1497–1550) the leading scientific publisher in Europe, who would go on ten years later to publish, Copernicus’ De revolutionibus, which was a high point in the astronomical revival.

All of this constitutes a clear indication of the steep rise in the demand for scientific instruments in the hundred years between 1450 and 1550; a demand that the metal workers of Nürnberg were more than happy to fill. In the period between Regiomontanus and the middle of the 16th century Nürnberg also became a home for some of the leading mathematici of the period, mathematicians, astronomers, astrologers, cartographers, instrument makers and globe makers almost certainly, like Regiomontanus, at least partially attracted to the city by the quality and availability of the scientific instruments.  Some of them are well known to historians of Renaissance science, Erhard Etzlaub, Johannes Werner, Johannes Stabius (not a resident but a frequent visitor), Georg Hartmann, Johannes Neudörffer and Johannes Schöner.**

There is no doubt that around 1500, Nürnberg was one of the major producers and exporters of scientific instruments and I hope that I have shown above, in what is little more than a sketch of a fairly complex process, that this owed very little to science but much to the general geo-political and economic developments of the first 500 years of the city’s existence.

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One of the most beautiful sets on instruments manufactured in Nürnberg late 16th century. Designed by Johannes Pretorius (1537–1616), professor for astronomy at the Nürnberger University of Altdorf and manufactured by the goldsmith Hans Epischofer (c. 1530–1585) Germanische National Museum

 

**for an extensive list of those working in astronomy, mathematics, instrument making in Nürnberg (542 entries) see the history section of the Astronomie in Nürnberg website, created by Dr Hans Gaab.

 

 

 

 

 

 

 

 

 

 

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Filed under Early Scientific Publishing, History of Astronomy, History of Cartography, History of Mathematics, History of Navigation, History of science, History of Technology, Renaissance Science

The emergence of modern astronomy – a complex mosaic: Part XXXII

In the seventeenth century large parts of Europe were still Catholic; in 1616 the Catholic Church had placed De revolutionibus and all other texts promoting a heliocentric world-view on the Index of Forbidden Books and in 1632 they added Galileo’s Dialogo sopra i due massimi sistemi del mondo (Dialogue Concerning the Two Chief World Systems), so the question arises, how was knowledge of the heliocentric model disseminated? The answer is, somewhat surprisingly, that the dissemination of the heliocentric hypothesis was, even in Catholic countries, widespread and through diverse channels.

First off, although De revolutionibus was placed on the Index in 1616, it was only placed there until corrected. In fact, somewhat against the norm, it was actually corrected surprisingly quickly and, with a few rather minor changes, became freely available again for Catholic scholars by 1621. The astronomers within the Church had been able to convince the theologians of the importance of Copernicus’ work as an astronomy book even if one rejected the truth of the heliocentric hypothesis. The only changes were that any statements of the factual truth of the hypothesis were removed, so anybody with a censured copy could quite happily think those statements back into place for himself.

The Lutheran Protestant Church also rejected the heliocentric hypothesis but never formally banned it in anyway. In fact, from very early on, the astronomers and mathematicians at the Lutheran universities had begun teaching Copernicus’ work as a purely mathematical, instrumentalist thesis, whilst rejecting it as a true account of the cosmos. It was used, for example, by Erasmus Reinhold (1511–1553) using Copernicus’ data and mathematical models to calculate the Prutenicae Tabulae (1551), without however committing to heliocentricity. They maintained this instrumentalist approach throughout the seventeenth century utilising the most up to date books as they became available, without crediting the hypothesis with any truth. From about 1630 onwards, Kepler’s Epitome Astronomiae Copernicanae (3 Vols. 1617–1621) and his Tabulae Rudolphinae (1627) became the leading textbooks for teaching the heliocentric hypothesis. The latter was used both sides of the religious divide because it was quite simply vastly superior in its accuracy to any other volume of planetary tables on the market.

However, the mainstream pro heliocentricity texts were not the only published sources spreading the information of the heliocentric hypothesis and making the information available across Europe. One, perhaps surprising, source was the yearly astrological almanacs.

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These annual pamphlets or booklets contained the astronomical and astrological data for the coming year, phases of the moon, hours of sunlight, any eclipse or planetary conjunctions etc. They also included basic horoscopes for the year covering political developments, weather forecasts, health issues and whatever. These were immensely popular and printed on cheap paper and not bound were reasonably cheap, so they sold in comparatively vast numbers, having much larger editions than any printed books. The market was fiercely contested so to make sure that their product was preferred by the potential customers, who came from all levels of society, the authors and/or publishers included editorials covering a wide range of topic. These editorials often contained medical issues but in the seventeenth century they also often contained popular expositions of the heliocentric hypothesis. Given the widespread consume of these publications it meant that basic knowledge of heliocentricity reached a large audience.

Another important source for the dissemination of the heliocentric hypothesis was in the writings of some of those who, nominally at least, opposed it. I will now take a brief look at two of those authors the Italian, Jesuit astronomer, mathematician and physicist Giovanni Battista Riccioli (1598–1671) and the French, priest, philosopher, astronomer and mathematician Pierre Gassendi (1592–1655) both of whom were highly influential and widely read scholars in the middle of the seventeenth century.

Pierre Gassendi is one of those figures in the history of science, who deserve to be better known than they are. Well known to historians of science and philosophy he remains largely unknown to those outside those disciplines. He was a central figure in the intellectual life of Europe in the middle of the seventeenth century part of the philosophical circle in Paris that included René Descartes, Marin Mersenne, Thomas Hobbes and Jean-Baptiste Morin amongst others. He also travelled to Holland and made the acquaintance of Isaac Beeckman. Probably his most important contribution to the evolution of science was his attempt to reconcile Epicurean atomism with Christian theology.

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Pierre Gassendi after Louis-Édouard Rioult. Source: Wikimedia Commons

Throughout his life he actively promoted the work of both Kepler and Galileo. He wrote and published a biography of Nicolas-Claude Fabri de Peiresc (1580–1637), his patron, an astronomer and another supporter of the works of Galileo.  Shortly before the end of his life he published a collective biography of Tycho Brahe, Nicolaus Copernicus, Georg von Peuerbach and Johannes Regiomontanus: Tychonis Brahei, equitis Dani, astronomorum Coryphaei, vita; accessit Nicolai Copernici; Georgii Peurbachii, et Joannis Regiomontani, astronomorum celebrium vita (1654).

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In 1645 Gassendi was appointed professor of mathematics at the Collège Royal in Paris and during his time there he wrote and published an astronomy textbook presenting both the Tychonic and heliocentric astronomical systems, Institvtio astronomica, iuxta hypothesis tam vetervm, qvam Copernici, et Tychonis. Dictata à Petro Gassendo … Eivsdem oratio inauguralis iteratò edita (1647).

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Although, as a Catholic priest, he presented the Tychonic system as the correct one his treatment of heliocentricity was detailed, thorough and very sympathetic. Perhaps somewhat too sympathetic, as it led to him being investigated by the Inquisition, who however gave him a clean bill of health. Because of his excellent reputation his book was read widely and acted as a major source for the dissemination of the heliocentric hypothesis.

Like Gassendi, Riccioli was an important and influential figure in seventeenth century science. From 1636 he was professor in Bologna where did much important work in astronomy and physics as well as being the teacher of Giovanni Domenico Cassini (1625–1712), who we will meet later in this series.

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Riccioli as portrayed in the 1742 Atlas Coelestis (plate 3) of Johann Gabriel Doppelmayer. Source: Wikimedia Commons

He is perhaps best known for his pioneering selenology together with his former student, Francesco Maria Grimaldi (1618–1663), which provided the nomenclature system for the moons geological features still in use today.  As stated earlier it was Riccioli, who provided the necessary empirical proof of Galileo’s laws of fall. He also hypothesised the existence of, what later became known as the Coriolis effect, if the Earth did in fact rotate.

If a ball is fired along a Meridian toward the pole (rather than toward the East or West), diurnal motion will cause the ball to be carried off [that is, the trajectory of the ball will be deflected], all things being equal: for on parallels of latitude nearer the poles, the ground moves more slowly, whereas on parallels nearer the equator, the ground moves more rapidly.

Having failed to detect it, it does exist but is too small to be measured using the methods available to Riccioli, he concluded that the Earth does not in fact rotate.

This was just one of many arguments pro and contra the heliocentric hypothesis that Riccioli presented in his Almagestum novum astronomiam veterem novamque complectens observationibus aliorum et propriis novisque theorematibus, problematibus ac tabulis promotam (Vol. I–III, 1651), a vast astronomical encyclopaedia that became a standard astronomical textbook throughout Europe. Although Riccioli rejected the heliocentric hypothesis his very detailed and thorough analysis of it with all its strengths and weaknesses meant that his book became a major source for those wishing to learn about it.

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Frontispiece of Riccioli’s 1651 New Almagest. Source: Wikimedia Commons

This famous frontispiece shows a semi-Tychonic system being weighed against a heliocentric system and being found more substantial. Ptolemaeus lies on the ground under the scales obviously defeated but he is saying “I will rise again”.

As we have seen, although not provable at that stage and nominally banned by the Catholic Church, information on and details of the heliocentric hypothesis were widespread and easily accessible throughout the seventeenth century from multiple sources and thus knowledge of it and interest in it continued to spread throughout the century.

 

 

 

 

 

 

 

 

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War, politics, religion and scientia

There is a strong tendency to view the history of science and the people who produced it in a sort of vacuum, outside of everyday society–Copernicus published this, Kepler published that, Newton synthesised it all… In fact the so-called scientific revolution took place in one of the most troubled times in European history, the age of the religious wars, the main one of which the Thirty Years War is thought to have been responsible directly and indirectly for the death of between one third and two thirds of the entire population of middle Europe. Far from being isolated from this turbulence the figures, who created modern science, were right in the middle of it and oft deeply involved and affected by it.

The idea for this blog post sort of crept into my brain as I was writing my review, two weeks ago, of two books about female spies during the English Revolution and Interregnum that is the 1640s to the 1660s. Isaac Newton was born during this period and grew up during it and, as I will now sketch, was personally involved in the political turbulence that followed on from it.

Born on Christmas Day in 1642 (os) shortly after the outbreak of the first of the three wars between the King and Parliament, Britain’s religious wars, he was just nine years old when Charles I was executed at the end of the second war.

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Portrait of Newton by Godfrey Kneller, 1689 Source: Wikimedia Commons

Newton was too young to be personally involved in the wars but others whose work would be important to his own later developments were. The Keplerian astronomer William Gascoigne (1612-1644), who invented the telescope micrometer, an important development in the history of the telescope, died serving in the royalist forces at the battle of Marston Moor. The mathematician John Wallis (1616–1703), whose Arithmetica Infinitorum (1656) strongly influenced Newton’s own work on infinite series and calculus, worked as a code breaker for Cromwelland later for Charles II after the restoration.

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John Wallis by Sir Godfrey Kneller

Newton first went up to university after the restoration but others of an earlier generation suffered loss of university position for being on the wrong side at the wrong time. John Wilkins (1614–1672), a parliamentarian and Cromwell’s brother-in-law, was appointed Master of Trinity College Cambridge, Newton’s college, in 1659 and removed from this position at the restoration. Wilkins’ Mathematical Magick (1648) had been a favourite of Newton’s in his youth.

Greenhill, John, c.1649-1676; John Wilkins (1614-1672), Warden (1648-1659)

Greenhill, John; John Wilkins (1614-1672), Warden (1648-1659); Wadham College, University of Oxford;

Newton’s political career began in 1689 following the so-called Glorious Revolution, when James II was chased out of Britain by William of Orange, his son-in-law, invited in by the parliament out of fear that James could reintroduce Catholicism into Britain. Newton sat in the House of Commons as MP for the University of Cambridge in the parliament of 1689, which passed the Bill of Rights, effectively a new constitution for England. Newton was not very active politically but he identified as a Whig, the party of his student Charles Montagu (1661–1715), who would go on to become one of the most powerful politicians of the age. It was Montagu, who had Newton appointed to lead the Royal Mint and it was also Montagu, who had Newton knighted in 1705in an attempt to get him re-elected to parliament.

In the standard version of story Newton represents the end of the scientific revolution and Copernicus (1473–1543) the beginning. Religion, politics and war all played a significant role in Copernicus’ life.

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Copernicus, the “Torun portrait” (anonymous, c. 1580), kept in Toruń town hall, Poland.

Copernicus spent the majority of his life living in the autonomous prince-bishopric of Warmia, where as a canon of the cathedral he was effectively a member of the government. Warmia was a Catholic enclave under the protection of the Catholic Crown of Poland but as the same time was geographically part of Royal Prussia ruled over by Duke Albrecht of Prussia (1490–1568), who had converted to Lutheran Protestantism in 1552. Ironically he was converted by Andreas Osiander (1498–1552), who would go on the author the controversial ad lectorum in Copernicus’ De revolutionibus. Relations between Poland and Royal Prussia were strained at best and sometimes spilled over into armed conflict. Between 1519 and 1521 there was a war between Poland and Royal Prussia, which took place mostly in Warmia. The Prussians besieged Frombork burning down the town, but not the cathedral, forcing Copernicus to move to Allenstein (Olsztyn), where he was put in charge of organising the defences during a siege from January to February 1521.  Military commander in a religious war in not a role usually associated with Copernicus. It is an interesting historical conundrum that, during this time of religious strife, De revolutionibus, the book of a Catholic cathedral canon, was published by a Protestant printer in a strongly Protestant city-state, Nürnberg.

The leading figure of the scientific revolution most affected by the religious wars of the age must be Johannes Kepler. A Lutheran Protestant he studied and graduated at Tübingen, one of the leading Protestant universities. However, he was despatched by the university authorities to become the mathematics teacher at the Protestant school in Graz in Styria, a deeply Catholic area in Austria in 1594. He was also appointed district mathematicus.

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Johannes Kepler Source: Wikimedia Commons

In 1598, Archduke Ferdinand, who became ruler of Styria in 1596, expelled all Protestant teachers and pastors from the province. Kepler was initially granted an exception because he had proved his worth as district mathematicus but in a second wave of expulsion, he too had to go. After failing to find employment elsewhere, he landed in Prague as an assistant to Tycho Brahe, the Imperial Mathematicus.

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Tycho Brahe Source: Wikimedia Commons

Once again he, like Tycho, was a Protestant in a Catholic city serving a Catholic Emperor, Rudolf II. Here he soon inherited Tycho’s position as Imperial Mathematicus. However, Rudolf was tolerant, more interested in Kepler’s abilities as an astrologer than in his religious beliefs. Apart from a substantial problem in getting paid in the permanently broke imperial court, Kepler now enjoyed a fairly quiet live for the next twelve years, then everything turned pear shaped once more.

In 1612, Rudolf’s younger brother Archduke Matthias deposed him and although Kepler was allowed to keep his title of Imperial Mathematicus, and theoretically at least, his salary but he was forced to leave Prague and become district mathematicus in Linz. In Linz Kepler, who openly propagated ecumenical ideas towards other Protestant communities, most notably the Calvinists, ran into conflict with the local Lutheran pastor. The pastor demanded that Kepler sign the Formula of Concord, basically a commitment to Lutheran theology and a rejection of all other theologies. Kepler refused and was barred from Holy Communion, a severe blow for the deeply religious astronomer. He appealed to the authorities in Tübingen but they up held the ban.

In 1618 the Thirty Years War broke out and in 1620 Linz was occupied by the Catholic army of Duke Maximilian of Bavaria, which caused problems for Kepler as a Lutheran. At the same time he was fighting for the freedom of his mother, Katharina, who had been accused of witchcraft. Although he won the court case against his mother, she died shortly after regaining her freedom. In 1625, the Counterreformation reach Linz and the Protestants living there were once again persecuted. Once more Kepler was granted an exception because of his status as Imperial Mathematicus but his library was confiscated making it almost impossible for him to work, so he left Linz.

Strangely, after two years of homeless wandering Kepler moved to Sagen in Silesia in 1628, the home of Albrecht von Wallenstein the commander of the Catholic forces in the war and for whom Kepler had interpreted a horoscope much earlier in life. Kepler never found peace or stability again in his life and died in Ulm in 1630. Given the turbulence in his life and the various forced moves, which took years rather than weeks, it is fairly amazing that he managed to publish eighty-three books and pamphlets between 1596 and his death in 1630.

A younger colleague of Kepler’s who also suffered during the Thirty Years’ War was Wilhelm Schickard, who Kepler had got to know during his time in Württemberg defending his mother. Schickard would go on to produce the illustrations both Kepler’s Epitome Astronomiae Copernicanae and his Harmonice Mundi, as well as inventing a calculating machine to help Kepler with his astronomical calculations. In 1632 Württemberg was invaded by the Catholic army, who brought the plague with them, by 1635 Schickard, his wife and his four living children, his sister and her three daughters had all died of the plague.

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Wilhelm Schickard, artist unknown Source: Wikimedia Commons

As I have pointed out on numerous occasions Galileo’s initial problems in 1615-16 had less to do with his scientific views than with his attempts to tell the theologians how to interpret the Bible, not an intelligent move at the height of the Counterreformation. Also in 1632 his problems were very definitely compounded by the fact that he was perceived to be on the Spanish side in the conflict between the Spanish and French Catholic authorities to influence, control the Pope, Urban VIII.

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Galileo Portrait by Ottavio Leoni Source: Wikimedia Commons

I will just mention in passing that René Descartes served as a soldier in the first two years of the Thirty Year’s War, at first in the Protestant Dutch States Army under Maurice of Nassau and then under the Catholic Duke of Bavaria, Maximilian. In 1620 he took part in the Battle of the White Mountain near Prague, which marked the end of Elector Palatine Frederick V’s reign as King of Bohemia. During his time in the Netherlands Descartes trained as a military engineer, which was his introduction to the works of Simon Stevin and Isaac Beeckman.

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René Descartes Portrait after Frans Hals Source: Wikimedia Commons

We have now gone full circle and are almost back to Isaac Newton. One interesting aspect of these troubled times is that although the problems caused by the wars, the religious disputes and the associated politics caused major problems in the lives of the astronomers and mathematicians, who were forced to live through them, and certainly affected their ability to carry on with their work, I can’t somehow imagine Copernicus working on De revolutionibus during the siege of Allenstein, the scholars themselves communicated quite happily across the religious divide.

Rheticus was treated as an honoured guest in Catholic Warmia although he was a professor at the University of Wittenberg, home to both Luther and Melanchthon. Copernicus himself was personal physician to both the Catholic Bishop of Frombork and the Protestant Duke of Royal Prussia. As we have seen, Kepler spent a large part of his life, although a devoted Protestant, serving high-ranking Catholic employers. The Jesuits, who knew Kepler from Prague, even invited him to take the chair for mathematics at the Catholic University of Bologna following the death of Giovanni Antonio Magini in 1617, assuring him that he did not need to convert. Although it was a very prestigious university Kepler, I think wisely, declined the invitation. The leading mathematicians of the time all communicated with each other, either directly or through intermediaries, irrespective of their religious beliefs. Athanasius Kircher, professor for mathematics and astronomy at the Jesuit Collegio Romano, collected astronomical data from Jesuits all over the world, which he then distributed to astronomers all over Europe, Catholic and Protestant, including for example the Lutheran Leibniz. Christiaan Huygens, a Dutch Calvinist, spent much of his life working as an honoured guest in Catholic Paris, where he met and influenced the Lutheran Leibniz.

When we consider the lives of scientists we should always bear in mind that they are first and foremost human beings, who live and work, like all other human beings, in the real world with all of its social, political and religious problems and that their lives are just as affected by those problems as everybody else.

 

 

 

 

 

 

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The emergence of modern astronomy – a complex mosaic: Part XXXI

Of all the discoveries made during the first phase of telescopic astronomical discoveries perhaps the more impactful was the discovery by various observers of the phases of Venus, which showed that Venus in fact orbited the Sun and not the Earth. This was the first real empirical proof that a pure geocentric system of astronomy was not possible. Because Mercury displayed the same behaviour as Venus in never moving more than a few degrees from the Sun as viewed from the Earth it was assumed, by analogy, that Mercury also orbited the Sun and not the Earth. This observed behaviour of Venus and Mercury had already led Martianus Capella (fl. c. 410–420) in late antiquity to hypothesise that both of them orbited the Sun and not the Earth. The Capellan system was widespread and well known in Europe during the Middle Ages. Unfortunately due to its comparatively small size and distance from the Earth, and the inadequacies of the most used Dutch or Galilean telescopes it would be almost three decades before anybody succeeded in observing the phases of Mercury.

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Naboth’s representation of Martianus Capella’s geo-heliocentric astronomical model (1573) Source: Wikimedia Commons

The knowledge that Venus orbited the sun and the assumption that Mercury also did led to the possibility of observing a so-called transit of one or other of them, i.e. the passage of the planet across the face of the Sun, as observed from the Earth. What would appear to be simply simple is in fact complicated by various factors. Principally the orbits of the Earth and Venus and/or Mercury around the Sun do not actually lie in the same plane. Taken Venus, for example, its orbit is tilted with respect to the Earth’s this means that most times when Venus is between the Earth and the Sun, it mostly passes above or below it rather than in front of it. Transits of Mercury take place approximately 13 or 14 times per century. Transits of Venus are less frequent taking place over a 243 year-cycle, two transits separated by eight years followed by long gaps of 121.5 and 105.5 years.

In order to be able to observe transits of the inner planets an astronomer requires detailed, accurate planetary tables, which allow them to predict the occurrence of the transit. Following the discovery of the phases of Venus it was Johannes Kepler (1571–1630), who, with the Rudolphine Tables (1627), first provided a set of planetary tables accurate enough to predict a transit of either Venus or Mercury.

According to Kepler’s calculations there should have been transits of both Mercury and Venus in 1631. Kepler did his best to draw astronomers’ attention to these occurrences with his De raris mirisque Anni 1631 including anadmonitio ad astronomos (1629), because their observation would they would help astronomers to settle the question of the true size of the planetary orbits, which up till then known relative to the Earth’s distance from the Sun, an unknown distance, and to determine the angular sizes of Mercury and Venus.

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Kepler died in 1630 but his son in law, the astronomer Jacob Bartsch (c. 1600–1633) published a printed pamphlet advising European astronomers of Kepler’s information.

Strangely there is no information that Bartsch observed the Transit of Mercury on 7 November 1631 but at least four other European astronomers did. The first was the German astronomer, astrologer, physician and calendar writer, Johann Rudrauf (1588–1654), known Johannes Remus Quietanus, a correspondent of Galileo, Kepler and Johannes Faber. Rudrauf observed the transit from Rufach in the Alsace border region between Germany and France, where he was town physician. The second was the Swiss Jesuit astronomer Johann Baptist Cysat (c. 1587–1657), who had been a student and assistant of Christoph Scheiner (c. 1573–1650) at the University of Ingolstadt and became his successor as professor of mathematics in 1618. Cysat observed the transit from Innsbruck in Austria.

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Johann Baptist Cysat Source: Wikimedia Commons

The third observer was the French philosopher, astronomer and mathematician, Pierre Gassendi (1592–1655), who observed the transit from Paris. A fourth unknown astronomer observed the transit from Ingolstadt.

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Pierre Gassendi after Louis-Édouard Rioult Source: Wikimedia Commons

Gassendi’s account of the transit was the most widely read and studied so he is usually credited with being the first to observe a transit of Mercury. On 6 December 1631 Gassendi tried to observe the transit of Venus that had been predicted by Kepler unaware that because it took place during the night it wasn’t visible from Europe.

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Transit of Mercury on May 9, 2016 Source: Wikimedia Commons

Kepler’s tables predicted no further transits of either Mercury or Venus for the seventeenth century so theoretically nobody would have set out to observe one. However, a young, self taught, English astronomer, Jeremiah Horrocks (1618–1641)did something quite extraordinary. Studying Kepler’s Rudolphine Tables Horrocks’ realised there would be another transit of Venus on 24 November 1639 (os) (4 December (ns)). Horrocks and his friend and fellow amateur astronomer William Crabtree (1610–1644) both observed the transit that Horrocks had predicted, Crabtree from Manchester and Horrocks from Much Hoole by Preston about 40 miles north of Manchester.

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Jeremiah Horrocks makes the first observation of the transit of Venus in 1639, as imagined by the artist W. R. Lavender in 1903 Source: Wikimedia Commons

Both of them made their observations by projecting the telescopic image on to a sheet of paper. Horrocks wrote an extensive report of their observations calculating both the size of Venus and the distance between the Earth and the Sun. Unfortunately Horrocks died in 1641 just 23 years old and his report of the first ever observation of a transit of Venus, Venus in sole visa, was first published was first published by the astronomer Johannes Hevelius (1611–1687)in Danzig in 1662.

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In 1639 the Italian Jesuit astronomer Giovanni Battista Zupi (1589–1650) succeeded in observing the orbital phases of Mercury using an astronomical telescope (two convex lenses) constructed by the Neapolitan astronomer, mathematician and telescope maker Francesco Fontana (1585–c. 1656).

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Zupi and Fontana often observed together and Zupi’s observations of the phases of Mercury were published in Fontana’s Novae Colestium in 1646.

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The observations of the transits of Mercury and Venus, and the phases of Mercury was final empirical proof, if it was needed, that both of the inner planets orbited the Sun and not the Earth nailing down the lid on the coffin of a pure geocentric, astronomical system. However, by the time these observations were made the majority of the astronomical community had already decided that only Kepler’s elliptical system or a Tychonic system with diurnal rotation were acceptable as the true system. The observations were of course compatible with both systems, so a final decision was not yet possible.

 

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It’s all a question of angles.

Thomas Paine (1736–1809) was an eighteenth-century political radical famous, or perhaps that should be infamous, for two political pamphlets, Common Sense (1776) and Rights of Man (1791) (he also wrote many others) and for being hounded out of England for his political views and taking part in both the French and American Revolutions.

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Thomas Paine portrait of Laurent Dabos c. 1792 Source: Wikimedia Commons

So I was more than somewhat surprised when Michael Brooks, author of the excellent The Quantum Astrologer’s Handbook, posted the following excerpt from Paine’s The Age of Reason, praising trigonometry as the soul of science:

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My first reaction to this beautiful quote was that he could be describing this blog, as the activities he names, astronomy, navigation, geometry, land surveying make up the core of the writings on here. This is not surprising as Ivor Grattan-Guinness in his single volume survey of the history of maths, The Rainbow of Mathematics: A History of the Mathematical Sciences, called the period from 1540 to 1660 (which is basically the second half of the European Renaissance) The Age of Trigonometry. This being the case I thought it might be time for a sketch of the history of trigonometry.

Trigonometry is the branch of mathematics that studies the relationships between the side lengths and the angles of triangles. Possibly the oldest trigonometrical function, although not regarded as part of the trigonometrical cannon till much later, was the tangent. The relationship between a gnomon (a fancy word for a stick stuck upright in the ground or anything similar) and the shadow it casts defines the angle of inclination of the sun in the heavens. This knowledge existed in all ancient cultures with a certain level of mathematical development and is reflected in the shadow box found on the reverse of many astrolabes.

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Shadow box in the middle of a drawing of the reverse of Astrolabium Masha’Allah Public Library Bruges [nl] Ms. 522. Basically the tangent and cotangent functions when combined with the alidade

Trigonometry as we know it begins with ancient Greek astronomers, in order to determine the relative distance between celestial objects. These distances were determined by the angle subtended between the two objects as observed from the earth. As the heavens were thought to be a sphere this was spherical trigonometry[1], as opposed to the trigonometry that we all learnt at school that is plane trigonometry. The earliest known trigonometrical tables were said to have been constructed by Hipparchus of Nicaea (c. 190–c. 120 BCE) and the angles were defined by chords of circles. Hipparchus’ table of chords no longer exist but those of Ptolemaeus (fl. 150 CE) in his Mathēmatikē Syntaxis (Almagest) still do.

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The chord of an angle subtends the arc of the angle. Source: Wikimedia Commons

With Greek astronomy, trigonometry moved from Greece to India, where the Hindu mathematicians halved the Greek chords and thus created the sine and also defined the cosine. The first recoded uses of theses function can be found in the Surya Siddhanta (late 4th or early 5th century CE) an astronomical text and the Aryabhatiya of Aryabhata (476–550 CE).

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Statue depicting Aryabhata on the grounds of IUCAA, Pune (although there is no historical record of his appearance). Source: Wikimedia Commons

Medieval Islam in its general acquisition of mathematical knowledge took over trigonometry from both Greek and Indian sources and it was here that trigonometry in the modern sense first took shape.  Muḥammad ibn Mūsā al-Khwārizmī (c. 780–c. 850), famous for having introduced algebra into Europe, produced accurate sine and cosine tables and the first table of tangents.

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Statue of al-Khwarizmi in front of the Faculty of Mathematics of Amirkabir University of Technology in Tehran Source: Wikimedia Commons

In 830 CE Ahmad ibn ‘Abdallah Habash Hasib Marwazi (766–died after 869) produced the first table of cotangents. Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī aṣ-Ṣābiʾ al-Battānī (c. 858–929) discovered the secant and cosecant and produced the first cosecant tables.

Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī (940–998) was the first mathematician to use all six trigonometrical functions.

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Abū al-Wafā Source: Wikimedia Commons

Islamic mathematicians extended the use of trigonometry from astronomy to cartography and surveying. Muhammad ibn Muhammad ibn al-Hasan al-Tūsī (1201–1274) is regarded as the first mathematician to present trigonometry as a mathematical discipline and not just a sub-discipline of astronomy.

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Iranian stamp for the 700th anniversary of Nasir al-Din Tusi’s death Source: Wikimedia Commons

Trigonometry came into Europe along with astronomy and mathematics as part the translation movement during the 11th and 12th centuries. Levi ben Gershon (1288–1344), a French Jewish mathematician/astronomer produced a trigonometrical text On Sines, Chords and Arcs in 1342. Trigonometry first really took off in Renaissance Europe with the translation of Ptolemaeus’ Geōgraphikḕ Hyphḗgēsis (Geographia) into Latin by Jacopo d’Angelo (before 1360–c. 1410) in 1406, which triggered a renaissance in cartography and astronomy.

The so-called first Viennese School of Mathematics made substantial contributions to the development of trigonometry in the sixteenth century. John of Gmunden (c. 1380–1442) produced a Tractatus de sinibus, chodis et arcubus. His successor, Georg von Peuerbach (1423–1461), published an abridgement of Gmunden’s work, Tractatus super propositiones Ptolemaei de sinibus et chordis together with a sine table produced by his pupil Regiomontanus (1436–1476) in 1541. He also calculated a monumental table of sines. Regiomontanus produced the first complete European account of all six trigonometrical functions as a separate mathematical discipline with his De Triangulis omnimodis (On Triangles) in 1464. To what extent his work borrowed from Arabic sources is the subject of discussion. Although Regiomontanus set up the first scientific publishing house in Nürnberg in 1471 he died before he could print De Triangulis. It was first edited by Johannes Schöner (1477–1547) and printed and published by Johannes Petreius (1497–1550) in Nürnberg in 1533.

At the request of Cardinal Bessarion, Peuerbach began the Epitoma in Almagestum Ptolomei in 1461 but died before he could complete it. It was completed by Regiomontanus and is a condensed and modernised version of Ptolemaeus’ Almagest. Peuerbach and Regiomontanus replaced the table of chords with trigonometrical tables and modernised many of the proofs with trigonometry. The Epitoma was published in Venice in 1496 and became the standard textbook for Ptolemaic geocentric astronomy throughout Europe for the next hundred years, spreading knowledge of trigonometry and its uses.

In 1533 in the third edition of the Apian/Frisius Cosmographia, Gemma Frisius (1508–1555) published as an appendix the first account of triangulationin his Libellus de locorum describendum ratione. This laid the trigonometry-based methodology of both surveying and cartography, which still exists today. Even GPS is based on triangulation.

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With the beginnings of deep-sea exploration in the fifteenth century first in Portugal and then in Spain the need for trigonometry in navigation started. Over the next centuries that need grew for determining latitude, for charting ships courses and for creating sea charts. This led to a rise in teaching trigonometry to seamen, as excellently described by Margaret Schotte in her Sailing School: Navigating Science and Skill, 1550–1800.

One of those students, who learnt their astronomy from the Epitoma was Nicolaus Copernicus (1473–1543). Modelled on the Almagest or more accurately the Epitoma, Copernicus’ De revolutionibus, published by Petreius in Nürnberg in 1543, also contained trigonometrical tables. WhenGeorg Joachim Rheticus (1514–1574) took Copernicus’ manuscript to Nürnberg to be printed, he also took the trigonometrical section home to Wittenberg, where he extended and improved it and published it under the title De lateribus et angulis triangulorum (On the Sides and Angles of Triangles) in 1542, a year before De revolutionibus was published. He would dedicate a large part of his future life to the science of trigonometry. In 1551 he published Canon doctrinae triangvlorvm in Leipzig. He then worked on what was intended to be the definitive work on trigonometry his Opus palatinum de triangulis, which he failed to finish before his death. It was completed by his student Valentin Otho (c. 1548–1603) and published in Neustadt an der Haardt in 1596.

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Source: Wikimedia Commons

In the meantime Bartholomäus Pitiscus (1561–1613) had published his own extensive work on both spherical and plane trigonometry, which coined the term trigonometry, Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuous in 1595.

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Source: Wikimedia Commons

This work was republished in expanded editions in 1600, 1608 and 1612. The tables contained in Pitiscus’ Trigonometria were calculated to five or six places, where as those of Rheticus were calculated up to more than twenty places for large angles and fifteenth for small ones. However, on inspection, despite the years of effort that Rheticus and Otho had invested in the work, some of the calculations were found to be defective. Pitiscus recalculated them and republished the work as Magnus canon doctrinae triangulorum in 1607. He published a second further improved version under the title Thesaurus mathematicus in 1613. These tables remained the definitive trigonometrical tables for three centuries only being replaced by Henri Andoyer’s tables in 1915–18.

We have come a long way from ancient Greece in the second century BCE to Germany at the turn of the seventeenth century CE by way of Early Medieval India and the Medieval Islamic Empire. During the seventeenth century the trigonometrical relationships, which I have up till now somewhat anachronistically referred to as functions became functions in the true meaning of the term and through analytical geometry received graphical presentations completely divorced from the triangle. However, I’m not going to follow these developments here. The above is merely a superficial sketch that does not cover the problems involved in actually calculating trigonometrical tables or the discovery and development of the various relationships between the trigonometrical functions such as the sine and cosine laws. For a detailed description of these developments from the beginnings up to Pitiscus I highly recommend Glen van Brummelen’s The Mathematics of the Heavens and the Earth: The Early History of Trigonometry, Princeton University Press, Princeton and Oxford, 2009.

 

[1] For a wonderful detailed description of spherical trigonometry and its history see Glen van Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, Princeton and Oxford, 2013

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Filed under History of Astronomy, History of Cartography, History of Mathematics, History of Navigation, History of science, Mediaeval Science, Renaissance Science

The emergence of modern astronomy – a complex mosaic: Part XXIX

One of the most well known popular stories told about Galileo is how he dropped balls from the Leaning Tower of Pisa to disprove the Aristotelian hypothesis that balls of different weights would fall at different speeds; the heavier ball falling faster. This event probably never happened but it is related as a prelude to his brilliant experiments with balls and inclined planes, which he carried out to determine empirically the correct laws of fall and which really did take place and for which he is justifiably renowned as an experimentalist. What is very rarely admitted is that the investigation of the laws of fall had had a several-hundred-year history before Galileo even considered the problem, a history of which Galileo was well aware.

We saw in the last episode that John Philoponus had actually criticised Aristotle’s concept of fall in the sixth century and had even carried out the ball drop experiment. However, unlike his impulse concept for projectile motion, which was taken up by Islamic scholars and passed on by them into the European Middle Ages, his correct criticism of Aristotle’s fall theory appears not to have been taken up by later thinkers.

As far as we know the first people, after Philoponus, to challenge Aristotle’s concept was the so-called Oxford Calculatores.

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Merton College in 1865 Source: Wikimedia Commons

This was a group of fourteenth-century, Aristotelian scholars at Merton College Oxford, who set about quantifying various theory of nature. These men–Thomas Bradwardine (c. 1300–1349), William of Heytesbury (c. 1313–c. 1372), Richard Swineshead (fl. c. 1340–1354) and John Dumbleton (c. 1310–c. 1349)–studied mechanics distinguishing between kinematics and dynamics, emphasising the former and investigating instantaneous velocity. They were the first to formulate the mean speed theorem, an achievement usually accredited to Galileo. The mean speed theorem states that a uniformly accelerated body, starting from rest, travels the same distance as a body with uniform speed, whose speed in half the final velocity of the accelerated body. The theory lies at the heart of the laws of fall.

The work of the Oxford Calculatores was quickly diffused throughout Europe and Nicole Oresme (c. 1320–1382), one of the so-called Parisian physicists,

Oresme

Portrait of Nicole Oresme: Miniature from Oresme’s Traité de l’espère, Bibliothèque Nationale, Paris, France, fonds français 565, fol. 1r. Source: Wikimedia Commons

and Giovanni di Casali (c. 1320–after 1374) both produced graphical representation of the theory.

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Oresme’s geometric verification of the Oxford Calculators’ Merton Rule of uniform acceleration, or mean speed theorem. Source: Wikimedia Commons

We saw in the last episode how Tartaglia applied mathematics to the problem of projectile motion and now we turn to a man, who for a time was a student of Tartaglia, Giambattista Benedetti (1530–1590). Like others before him Bendetti turned his attention to Aristotle’s concept of fall and wrote and published in total three works on the subject that went a long way towards the theory that Galileo would eventually publish. In his Resolutio omnium Euclidis problematum (1553) and his Demonstratio proportionum motuum localium (1554) he argued that speed is dependent not on weight but specific gravity and that two objects of the same material but different weights would fall at the same speed.

800px-Benedetti_-_Resolutio_omnium_Euclidis_problematum,_1553_-_11183

Source: Wikimedia Commons

However, in a vacuum, objects of different material would fall at different speed. Benedetti brought an early version of the thought experiment, usually attributed to Galileo, of viewing two bodies falling separately or conjoined, in his case by a cord.  Galileo considered a roof tile falling complete and then broken into two.

In a second edition of the Demonstratio (1554) he addressed surface area and resistance of the medium through which the objects are falling. He repeated his theories in his Demonstratio proportionum motuum localium (1554), where he explains his theories with respect to the theory of impetus. We know that Galileo had read his Benedetti and his own initial theories on the topic, in his unpublished De Motu, were very similar.

In the newly established United Provinces (The Netherlands) Simon Stevin (1548–1620) carried out a lot of work applying mathematics to various areas of physics. However in our contexts more interesting were his experiments in 1586, where he actually dropped lead balls of different weights from the thirty-foot-high church tower in Delft and determined empirically that they fell at the same speed, arriving at the ground at the same time.

Simon-stevin

Source: Wikimedia Commons

Some people think that because Stevin only wrote and published in Dutch that his mathematical physics remained largely unknown. However, his complete works published initially in Dutch were translated into both French and Latin, the latter translation being carried out by Willebrord Snell. As a result his work was well known in France, the major centre for mathematical physics in the seventeenth century.

In Italy the Dominican priest Domingo de Soto (1494–1560) correctly stated that a body falls with a constant, uniform acceleration. In his Opus novum, De Proportionibus numerorum, motuum, ponderum, sonorum, aliarumque rerum mensurandarum. Item de aliza regula (1570) Gerolamo Cardano (1501–1576) demonstrates that two balls of different sizes will fall from a great height in the same time. The humanist poet and historian, Benedetto Varchi (c. 1502–1565) in 1544 and Giuseppe Moletti (1531–1588), Galileo’s predecessor as professor of mathematics in Padua, in 1576 both reported that bodies of different weights fall at the same speed in contradiction to Aristotle, as did Jacopo Mazzoni (1548–1598), a philosopher at Padua and friend of Galileo, in 1597. However none of them explained how they arrived at their conclusions.

Of particular relevance to Galileo is the De motu gravium et levium of Girolamo Borro (1512–1592) one of Galileo’s teachers in Pisa. In a dispute with his colleague Francesco Buonamici (1533–1603), another Pisan professor, Borro carried out experiments in which he threw objects of different material and the same weights out of a high window to test Aristotle’s theory, which he describes in his book. Borro’s work is known to have had a strong influence on Galileo’s early work in this area.

When Galileo started his own extensive investigations into the problem of fall in the late sixteenth century he was tapping into an extensive stream of previous work on the subject of which he was well aware and which to some extent had already done much of the heavy lifting. This raises the question as to what extent Galileo deserves his reputation as the man, who solved the problem of fall.

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Galileo Portrait by Ottavio Leoni Source: Wikimedia Commons

We saw in the last episode that his much praised Dialogo, his magnum opus on the heliocentricity contra geocentricity debate, not only contributed nothing new of substance to that debate but because of his insistence on retaining the Platonic axioms, his total rejection of the work of both Tycho Brahe and Kepler and his rejection of the strong empirical evidence for the supralunar nature of comets he actually lagged far behind the current developments in that debate. The result was that the Dialogo could be regarded as superfluous to the astronomical system debate. Can the same be said of the contribution of the Discorsi e dimostrazioni matematiche intorno a due nuove scienze (Discourses and Mathematical Demonstrations Relating to Two New Sciences) (1638) to the debate on motion? The categorical answer is no; the Discorsi is a very important contribution to that debate and Galileo deserves his reputation as a mathematical physicist that this book gave him.

Galileo_Galilei,_Discorsi_e_Dimostrazioni_Matematiche_Intorno_a_Due_Nuove_Scienze,_1638_(1400x1400)

Source: Wikimedia Commons

What did Galileo contribute to the debate that was new? It not so much that he contributed much new to the debate but that he gave the debate the solid empirical and mathematical foundation, which it had lacked up till this point. Dropping weights from heights to examine the laws of fall suffers from various problems. It is extremely difficult to ensure that the object are both released at the same time, it is equally difficult to determine if they actually hit the ground at the same time and the whole process is so fast, that given the possibilities available at the time, it was impossible to measure the time taken for the fall. All of the previous experiments of Stevin et al were at best approximations and not really empirical proofs in a strict scientific sense. Galileo supplied the necessary empirical certainty.

Galileo didn’t drop balls he rolled them down a smooth, wooden channel in an inclined plane that had been oiled to remove friction. He argued by analogy the results that he achieved by slowing down the acceleration by using an inclined plane were equivalent to those that would be obtained by dropping the balls vertically. Argument by analogy is of course not strict scientific proof but is an often used part of the scientific method that has often, as in this case, led to important new discoveries and results.  He released one ball at a time and timed them separately thus eliminating the synchronicity problem. Also, he was able with a water clock to time the balls with sufficient accuracy to make the necessary mathematical calculations. He put the laws of falls on a sound empirical and mathematical footing. One should also not neglect the fact that Galileo’s undoubtable talent as a polemicist made the content of the Discorsi available in a way that was far more accessible than anything that had preceded it.

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Galileo’s demonstration of the law of the space traversed in case of uniformly varied motion. It is the same demonstration that Oresme had made centuries earlier. Source: Wikimedia Commons

For those, who like to believe that Catholics and especially the Jesuits were anti-science in the seventeenth century, and unfortunately they still exist, the experimental confirmation of Galileo’s law of fall, using direct drop rather than an inclined plane, was the Jesuit, Giovanni Battista Riccioli(1598–1671).

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Giovanni Battista Riccioli Source: Wikimedia Commons

The Discorsi also contains Galileo’s work on projectile motion, which again was important and influential. The major thing is the parabola law that states that anything projected upwards and away follows a parabolic path. Galileo was not the only natural philosopher, who determined this. The Englishman Thomas Harriot (c. 1560–1621) also discovered the parabola law and in fact his work on projectile motion went well beyond that of Galileo. Unfortunately, he never published anything so his work remained unknown.  One of Galileo’s acolytes, Bonaventura Cavalieri (1598–1647),

Bonaventura_Cavalieri

Bonaventura Cavalieri Source: Wikimedia Commons

was actually the first to publish the parabola law in his Lo Specchio Ustorio, overo, Trattato delle settioni coniche (The Burning Mirror, or a Treatise on Conic Sections) 1632.

This brought an accusation of intellectual theft from Galileo and it is impossible to tell from the ensuing correspondence, whether Cavalieri discovered the law independently or borrowed it without acknowledgement from Galileo.

The only problem that remained was what exactly was impetus. What was imparted to bodies to keep them moving? The answer was nothing. The solution was to invert the question and to consider what makes moving bodies cease to move? The answer is if nothing does, they don’t. This is known as the principle of inertia, which states that a body remains at rest or continues to move in a straight line unless acted upon by a force. Of course, in the early seventeenth century nobody really knew what force was but they still managed to discover the basic principle of inertia. Galileo sort of got halfway there. Still under the influence of the Platonic axioms, with their uniform circular motion, he argued that a homogenous sphere turning around its centre of gravity at the earth’s surface forever were there no friction at its bearings or against the air. Because of this Galileo is often credited with provided the theory of inertia as later expounded by Newton but this is false.

The Dutch scholar Isaac Beeckman (1588–1637) developed the concept of rectilinear inertia, as later used by Newton but also believed, like Galileo, in the conservation of constant circular velocity. Beeckman is interesting because he never published anything and his writing only became known at the beginning of the twentieth century. However, Beeckman was in contact, both personally and by correspondence, with the leading French mathematicians of the period, Descartes, Gassendi and Mersenne. For a time he was Descartes teacher and much of Descartes physics goes back to Beeckman. Descartes learnt the principle of inertia from Beeckman and it was he who published and it was his writings that were Newton’s source. The transmission of Beeckman’s work is an excellent illustration that scientific information does not just flow over published works but also through personal, private channels, when scientists communicate with each other.

With the laws of fall, the parabola law and the principle of inertia the investigators in the early seventeenth century had a new foundation for terrestrial mechanics to replace those of Aristotle.

 

 

 

 

 

 

 

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