Category Archives: Renaissance Science

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.

Astrolabium_Masha'allah_Public_Library_Brugge_Ms._522.tif

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.

Rheticus_Opus_Palatinum_De_Triangulis

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,

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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.

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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.

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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),

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

One of the central problems in the transition from the traditional geocentric astronomy/cosmology to a heliocentric one was that the system that the Early Modern astronomers inherited from their medieval predecessors was not just an uneasy amalgam of Aristotelian cosmology and Ptolemaic astronomy but it also included Aristotle’s (384–322 BCE) theories of terrestrial and celestial motion all tied together in a complete package. Aristotle’s theory of motion was part of his more general theory of change and differentiated between natural motion and unnatural or violent motion.

The celestial realm in Aristotle’s cosmology was immutable, unchanging, and the only form of motion was natural motion that consisted of uniform, circular motion; a theory that he inherited from Plato (c. 425 – c.347 BCE), who in turn had adopted it from Empedocles (c. 494–c. 434 BCE).

His theory of terrestrial motion had both natural and unnatural motion. Natural motion was perpendicular to the Earth’s surface, i.e. when something falls to the ground. Aristotle explained this as a form of attraction, the falling object returning to its natural place. Aristotle also claimed that the elapsed time of a falling body was inversely proportional to its weight. That is, the heavier an object the faster it falls. All other forms of motion were unnatural. Aristotle believed that things only moved when something moved them, people pushing things, draught animals pulling things. As soon as the pushing or pulling ceased so did the motion.  This produced a major problem in Aristotle’s theory when it came to projectiles. According to his theory when a stone left the throwers hand or the arrow the bowstring they should automatically fall to the ground but of course they don’t. Aristotle explained this apparent contradiction away by saying that the projectile parted the air through which it travelled, which moved round behind the projectile and pushed it further. It didn’t need a philosopher to note the weakness of this more than somewhat ad hoc theory.

If one took away Aristotle’s cosmology and Ptolemaeus’ astronomy from the complete package then one also had to supply new theories of celestial and terrestrial motion to replace those of Aristotle. This constituted a large part of the development of the new physics that took place during the so-called scientific revolution. In what follows we will trace the development of a new theory, or better-said theories, of terrestrial motion that actually began in late antiquity and proceeded all the way up to Isaac Newton’s (1642–1726) masterpiece Principia Mathematica in 1687.

The first person to challenge Aristotle’s theories of terrestrial motion was John Philoponus (c. 490–c. 570 CE). He rejected Aristotle’s theory of projectile motion and introduced the theory of impetus to replace it. In the impetus theory the projector imparts impetus to the projected object, which is used up during its flight and when the impetus is exhausted the projectile falls to the ground. As we will see this theory was passed on down to the seventeenth century. Philoponus also rejected Aristotle’s quantitative theory of falling bodies by apparently carrying out the simple experiment usually attributed erroneously to Galileo, dropping two objects of different weights simultaneously from the same height:

but this [view of Aristotle] is completely erroneous, and our view may be completely corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights, one many times heavier than the other you will see that the ratio of the times required for the motion does not depend [solely] on the weights, but that the difference in time is very small. …

Philoponus also removed Aristotle’s distinction between celestial and terrestrial motion in that he attributed impetus to the motion of the planets. However, it was mainly his terrestrial theory of impetus that was picked up by his successors.

In the Islamic Empire, Ibn Sina (c. 980–1037), known in Latin as Avicenne, and Abu’l-Barakāt Hibat Allah ibn Malkā al-Baghdādī (c. 1080–1164) modified the theory of impetus in the eleventh century.

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Avicenne Portrait (1271) Source: Wikimedia Commons

Nur ad-Din al-Bitruji (died c. 1204) elaborated it at the end of the twelfth century. Like Philoponus, al-Bitruji thought that impetus played a role in the motion of the planets.

 

Brought into European thought during the scientific Renaissance of the twelfth and thirteenth centuries by the translators it was developed by Jean Buridan  (c. 1301–c. 1360), who gave it the name impetus in the fourteenth century:

When a mover sets a body in motion he implants into it a certain impetus, that is, a certain force enabling a body to move in the direction in which the mover starts it, be it upwards, downwards, sidewards, or in a circle. The implanted impetus increases in the same ratio as the velocity. It is because of this impetus that a stone moves on after the thrower has ceased moving it. But because of the resistance of the air (and also because of the gravity of the stone) which strives to move it in the opposite direction to the motion caused by the impetus, the latter will weaken all the time. Therefore the motion of the stone will be gradually slower, and finally the impetus is so diminished or destroyed that the gravity of the stone prevails and moves the stone towards its natural place. In my opinion one can accept this explanation because the other explanations prove to be false whereas all phenomena agree with this one.

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Jean Buridan Source

The impetus theory was now a part of medieval Aristotelian natural philosophy, which as Edward Grant pointed out was not Aristotle’s natural philosophy.

In the sixteenth century the self taught Italian mathematician Niccolò Fontana (c. 1500–1557), better known by his nickname, Tartaglia, who is best known for his dispute with Cardanoover the general solution of the cubic equation.

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Niccolò Fontana Tartaglia Source: Wikimedia Commons

published the first mathematical analysis of ballistics his, Nova scientia (1537).

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His theory of projectile trajectories was wrong because he was still using the impetus theory.

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However, he was the first to demonstrate that an angle of 45° for a canon gives the widest range.

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His book was massively influential in the sixteenth century and it also influenced Galileo, who owned a heavily annotated copy of the book.

We have traced the path of the impetus theory from its inception by John Philoponus up to the second half of the sixteenth century. Unlike the impetus theory Philoponus’ criticism of Aristotle’s theory of falling bodies was not taken up directly by his successors. However, in the High Middle Ages Aristotelian scholars in Europe did begin to challenge and question exactly those theories and we shall be looking at that development in the next section.

 

 

 

 

 

 

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Filed under History of Islamic Science, History of Mathematics, History of Physics, Mediaeval Science, Renaissance Science

Stylish writing is not necessarily good science

I have become somewhat infamous for writing #histSTM blog posts that are a predominately negative take on the scientific achievements of Galileo Galilei. In fact I think I probably made my breakthrough as a #histsci blogger with my notorious Extracting the Stopper post, deflating Galileo’s popular reputation. I actually got commissioned to write a toned down version of that post for AEON several years later. In my opinion Galileo was an important figure in the evolution of science during the early seventeenth century but his reputation has been blown up out of all proportion, well beyond his actual contributions. To make a simple comparison, in the same period of time the contributions of Johannes Kepler were immensely greater and more significant than those made by Galileo but whereas Galileo is regarded as one of the giants of modern science and is probably one of the three most well known historical practitioners of the mathematical sciences, alongside Newton and Einstein, Kepler is at best an also ran, whose popular image is not even a fraction of that of Galileo’s. This of course raises the question, why? What does/did Galileo have that Kepler didn’t? I think the answer lies in Galileo’s undeniable talents as a writer.

Galileo was a master stylist, a brilliant polemicist and science communicator, whose major works are still a stimulating pleasure to read. If you ask people about Galileo they will more often than not quote one of his well-known turns of phrase rather than his scientific achievements. The two books trope with its ‘mathematics is the language of nature’, which in the original actually reads: 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. Or the much-loved, the Bible shows the way to go to heaven, not the way the heavens go, which again in the original reads: The intention of the Holy Ghost is to teach us how one goes to heaven, not how heaven goes. It is a trivial truth that Galileo had a way with words.

This cannot be said of Johannes Kepler. I shall probably bring the wrath of a horde of Kepler scholars on my head for saying this but even in translation, Johannes Kepler is anything but an easy read. Galileo even commented on this. When confronted with Kepler’s Dioptrice (1611), one of the most important books on optics ever written, Galileo complained that it was turgid and unreadable. Having ploughed my way through it in German translation, I sympathise with Galileo’s negative judgement. However, in his rejection Galileo failed to realise just how scientifically important the Dioptrice actually was. Nobody in their right mind would describe Kepler as a master stylist or a brilliant polemicist.

I think this contrast in literary abilities goes a long way to explaining the very different popular conceptions of the two men. People read Galileo’s major works or selections from them and are stimulated and impressed by his literary mastery, whereas Kepler’s major works are not even presented, as something to be read by anyone, who is not a historian of science. One just gets his three laws of planetary motion served up in modern guise, as a horribly mathematical side product of heliocentricity.

Of course, a serious factor in their respective notorieties is Galileo’s infamous trial by the Roman Inquisition. This was used to style him as a martyr for science, a process that only really began at the end of the eighteenth and beginning of the nineteenth centuries. Kepler’s life, which in many ways was far more spectacular and far more tragic than Galileo’s doesn’t have such a singular defining moment in it.

Returning to the literary theme I think that what has happened is that non-scientists and non-historians of science have read Galileo and impressed by his literary abilities, his skill at turning a phrase, his adroit, and oft deceitful, presentation of pro and contra arguments often fail to notice that they are being sold a pup. As I tried to make clear in the last episode of my continuing ‘the emergence of modern astronomy’ series although Galileo’s Dialogo has an awesome reputation in Early Modern history, its scientific value is, to put it mildly, negligible. To say this appears to most people as some form of sacrilege, “but the Dialogo is an important defence of science against the forces of religious ignorance” or some such they would splutter. But in reality it isn’t, as I hope I made clear the work contributed nothing new to the on going debate and all that Galileo succeeded in doing was getting up the Pope’s nose.

The same can be said of Il Saggiatore, another highly praised work of literature. As I commented in another post the, oft quoted line on mathematics, which had led to Galileo being praised as the man, who, apparently single handed, mathematized the physical science was actually, when he wrote it, old hat and others had been writing the book of nature in the language of mathematics for at least one hundred years before Galileo put pen to paper but none of them had taken the time to express what they were doing poetically. In fact it took historians of science a long time to correct this mistaken perception, as they also tended to suffer from a serious dose of Galileo adoration. The core of Il Saggiatore is as I have explained elsewhere is total rubbish, as Galileo is arguing against the scientific knowledge of his time with very spurious assertions merely so that he doesn’t have to acknowledge that Grassi is right and he is wrong. An admission that very few Galileo scholars are prepared to make in public, it might tarnish his reputation.

Interestingly one work that deserves its historical reputation Galileo’s Sidereus Nuncius, also suffers from serious scientific deficits that tend to get overlooked. Written and published in haste to avoid getting beaten to the punch by a potential, unknown rival the book actually reads more like an extended press release that a work of science. It might well be that Galileo intended to write a more scientific evaluation of his telescopic observations and discoveries once he had established his priority but somehow, having become something of a scientific superstar overnight, he never quite got round to it. This is once again a failing that most readers tend to overlook, over awed by the very impressive literary presentation.

Much of Galileo’s written work is actually more appearance than substance, or as the Germans say Mehr Schein als Sein, but ironically, there is one major work of Galileo’s that is both literarily brilliant and scientifically solid but which tends to get mostly overlooked, his Discorsi. The experiments on which part of it is based get described by the book itself remains for most people largely unknown. I shall be taking a closer look at it in a later post.

 

 

 

 

 

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

Without a doubt the most well-known, in fact notorious, episode in the transition from a geocentric to a heliocentric cosmology/astronomy in the seventeenth century was the publication of Galileo Galilei’s Dialogo sopra i due massimi sistemi del mondo (Dialogue Concerning the Two Chief World Systems) in 1632 and his subsequent trial and conviction by the Supreme Sacred Congregation of the Roman and Universal Inquisition or simply Roman Inquisition; an episode that has been blown up out of all proportions over the centuries. It would require a whole book of its own to really do this subject justice but I shall deal with it here in two sketches. The first to outline how and why the publication of this book led to Galileo’s trial and the second to assess the impact of the book on the seventeenth century astronomical/cosmological debate, which was much less than is often claimed.

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Frontispiece and title page of the Dialogo, 1632 Source: Wikimedia Commons

The first salient point is Galileo’s social status in the early seventeenth century. Nowadays we place ‘great scientists’ on a pedestal and accord them a very high social status but this was not always the case. In the Renaissance, within society in general, natural philosophers and mathematicians had a comparatively low status and within the ruling political and religious hierarchies Galileo was effectively a nobody. Yes, he was famous for his telescopic discoveries but this did not change the fact that he was a mere mathematicus. As court mathematicus and philosophicus to the Medici in Florence he was little more than a high-level court jester, he should reflect positively on his masters. His role was to entertain the grand duke and his guests at banquets and other social occasions with his sparkling wit, either in the form of a discourse or if a suitable opponent was at hand, in a staged dispute. Points were awarded not for truth content but for verbal brilliance. Galileo was a master at such games. However, his real status as a courtier was very low and should he bring negative attention to the court, they would drop him without a thought, as they did when the Inquisition moved against him.

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Galileo Portrait by Ottavio Leoni Source: Wikimedia Commons

As a cardinal, Maffeo Barberini (1568–1644) had befriended Galileo when his first came to prominence in 1611 and he was also an admirer of the Accademia dei Lincei. When he was elected Pope in 1623 the Accademia celebrated his election and amongst other things presented him with a copy of Galileo’s Il Saggiatore, which he read and apparently very much enjoyed. As a result he granted Galileo several private audiences, a great honour. Through his actions Barberini had raised Galileo to the status of papal favourite, a situation not without its dangers.

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C. 1598 painting of Maffeo Barberini at age 30 by Caravaggio Source: Wikimedia Commons

Mario Biagioli presents the, I think correct, hypothesis that having raised Galileo up as a court favourite Barberini then destroyed him. Such behaviour was quite common under absolutist rulers, as a power demonstration to intimidate potential rebels. Galileo was a perfect victim for such a demonstration highly prominent and popular but with no real political or religious significance. Would Barberini have staged such a demonstration at the time? There is evidence that he was growing more and more paranoid during this period. Barberini, who believed deeply in astrology, heard rumours that an astrologer had foreseen his death in the stars. His death was to coincide with a solar eclipse in 1630. Barberini with the help of his court astrologer, Tommaso Campanella (1568–1639) took extreme evasive action and survived the cosmic threat but he had Orazio Morandi (c. 1570–1630), a close friend and supporter of Galileo’s, arrested and thrown in the papal dungeons, where he died, for having cast the offending horoscope.

Turning to the Dialogo, the official bone of contention, Galileo succeeded in his egotism in alienating Barberini with its publication. Apparently during the phase when he was very much in Barberini’s good books, Galileo had told the Pope that the Protestants were laughing at the Catholics because they didn’t understand the heliocentric hypothesis. Of course, during the Thirty Years War any such mockery was totally unacceptable. Barberini gave Galileo permission to write a book presenting and contrasting the heliocentric and geocentric systems but with certain conditions. Both systems were to be presented as equals with no attempts to prove the superiority or truth of either and Galileo was to include the philosophical and theological opinion of the Pope that whatever the empirical evidence might suggest, God in his infinite wisdom could create the cosmos in what ever way he chose.

The book that Galileo wrote in no way fulfilled the condition stated by Barberini. Presented as a discussion over four days between on the one side a Copernican, Salviati named after Filippo Salviati (1682–1614) a close friend of Galileo’s and Sagredo, supposedly neutral but leaning strongly to heliocentricity, named after Giovanni Francesco Sagredo (1571–1620) another close friend of Galileo’s. Opposing these learned gentlemen is Simplicio, an Aristotelian, named after Simplicius of Cilicia a sixth-century commentator on Aristotle. This name is with relative certainty a play on the Italian word “semplice”, which means simple as in simple minded. Galileo stacked the deck from the beginning.

The first three days of discussion are a rehash of the previous decades of discoveries and developments in astronomy and cosmology with the arguments for heliocentricity, or rather against geocentricity in its Ptolemaic/Aristotelian form, presented in their best light and the counter arguments presented decidedly less well. Galileo was leaving nothing to chance, he knew who was going to win this discussion. The whole thing is crowned with Galileo’s theory of the tides on day four, which he falsely believed, despite its very obvious flaws, to be a solid empirical proof of the Earth’s movements in a heliocentric model. This was in no way an unbiased presentation of two equal systems but an obvious propaganda text for heliocentricity. Worse than this, he placed the Pope’s words on the subject in the mouth of Simplicio, the simpleton, not a smart move. When it was published the shit hit the fan.

However, before considering the events leading up to the trial and the trial itself there are a couple of other factors that prejudiced the case against Galileo. In order to get published at all, the book, as with every other book, had to be given publication permission by the censor. To repeat something that people tend to forget, censorship was practiced by all secular and all religious authorities throughout the whole of Europe and was not peculiar to the Catholic Church. Freedom of speech and freedom of thought were alien concepts in the world of seventeenth century religion and politics. Galileo wanted initially to title the book, Dialogue on the Ebb and Flow of the Seas, referring of course to his theory of the tides, and include a preface to this effect. He was told to remove both by the censor, as they, of course, implied a proof of heliocentricity. Because of an outbreak of the plague, Galileo retired to Florence to write his book and preceded to play the censor in Florence and the censor in Rome off against each other, which meant that the book was published without being properly controlled by a censor. This, of course, all came out after publication and did not help Galileo’s case at all; he had been far too clever for his own good.

Another major problem had specifically nothing to do with Galileo in the first instance but rebounded on him at the worst time.  On 8 March 1632 Cardinal Borgia castigated the Pope for not supporting King Philipp IV of Spain against the German Protestants. The situation almost degenerated into a punch up with the Swiss Guard being called in to separate the adversaries. As a result Barberini decided to purge the Vatican of pro-Spanish elements. One of the most prominent men to be banished was Giovanni Ciampoli (1589–1643) Barberini’s chamberlain. Ciampoli was an old friend and supporter of Galileo and a member of the Accademia dei Lincei. He was highly active in helping Galileo trick the censors and had read the manuscript of the Dialogo, telling Barberini that it fulfilled his conditions. His banishment was a major disaster for Galileo.

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Giovanni Ciampoli Source: Wikimedia Commons

One should of course also not forget that Galileo had effectively destroyed any hope of support from the Jesuits, the leading astronomers and mathematicians of the age, who had very actively supported him in 1611, with his unwarranted and libellous attacks on Grazi and Scheiner in his Il Saggiatore. He repeated the attacks on Scheiner in the Dialogo, whilst at the same time plagiarising him, claiming some of Scheiner’s sunspot discoveries as his own. There is even some evidence that the Jesuits worked behind the scenes urging the Pope to put Galileo on trial.

When the Dialogo was published it immediately caused a major stir. Barberini appointed officials to read and assess it. Their judgement was conclusive, the Dialogo obviously breached the judgement of 1616 forbidding the teaching of heliocentricity as a factual theory. Anybody reading the Dialogo today would confirm that judgement. The consequence was that Galileo was summoned to Rome to answer to the Inquisition. Galileo stalled claiming bad health but was informed either he comes or he would be fetched. The Medici’s refused to support him; they did no consider him worth going into confrontation with the Pope for.

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Ferdinando II de’ Medici Grand Duke of Tuscany in Coronation Robes (school of Justus Sustermans). Source: Wikimedia Commons

We don’t need to go into details of the trial. Like all authoritarian courts the Inquisition didn’t wish to try their accused but preferred them to confess, this was the case with Galileo. During his interviews with the Inquisition Galileo was treated with care and consideration because of his age and bad health. He was provided with an apartment in the Inquisition building with servants to care for him. At first he denied the charges but when he realised that this wouldn’t work he said that he had got carried away whilst writing and he offered to rewrite the book. This also didn’t work, the book was already on the market and was a comparative best seller, there was no going back. Galileo thought he possessed a get out of jail free card. In 1616, after he had been interviewed by Bellarmino, rumours circulated that he had been formally censured by the Inquisition. Galileo wrote to Bellarmino complaining and the Cardinal provided him with a letter stating categorically that this was not the case. Galileo now produced this letter thinking it would absolve him of the charges. The Inquisition now produced the written version of the statement that had been read to Galileo by an official of the Inquisition immediately following his interview with Bellarmino expressly forbidding the teaching of the heliocentric theory as fact. This document still exists and there have been discussions as to its genuineness but the general consensus is that it is genuine and not a forgery. Galileo was finished, guilty as charged. Some opponents of the Church make a lot of noise about Galileo being shown the instruments of torture but this was a mere formality in a heresy trial and at no point was Galileo threatened with torture.

The rest is history. Galileo confessed and formally adjured to the charge of grave suspicion of heresy, compared to heresy a comparatively minor charge. He was sentenced to prison, which was immediately commuted to house arrest. He spent the first months of his house arrest as the guest of Ascanio II Piccolomini (1590–1671), Archbishop of Siena,

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Ascanio II Piccolomini Source: Wikimedia Commons

until Barberini intervened and sent him home to his villa in Arcetri. Here he lived out his last decade in comparative comfort, cared for by loyal servants, receiving visitor and writing his most important book, Discorsi e dimostrazioni matematiche intorno a due nuove scienze (Discourses and Mathematical Demonstrations Relating to Two New Sciences).

Galileo’s real crime was hubris, trying to play an absolutist ruler, the Pope, for a fool. Others were executed for less in the seventeenth century and not just by the Catholic Church. Galileo got off comparatively lightly.

What role did the Dialogo actually play in the ongoing cosmological/astronomical debate in the seventeenth century? The real answer is, given its reputation, surprisingly little. In reality Galileo was totally out of step with the actual debate that was taking place around 1630. Driven by his egotistical desire to be the man, who proved the truth of heliocentricity, he deliberately turned a blind eye to the most important developments and so side lined himself.

We saw earlier that around 1613 there were more that a half a dozen systems vying for a place in the debate, however by 1630 nearly all of the systems had been eliminated leaving just two in serious consideration. Galileo called his book Dialogue Concerning the Two Chief World Systems, but the two systems that he chose to discuss, the Ptolemaic/Aristotelian geocentric system and the Copernican heliocentric system, were ones that had already been rejected by almost all participants in the debate by 1630 . The choice of the pure geocentric system of Ptolemaeus was particularly disingenuous, as Galileo had helped to show that it was no longer viable twenty years earlier. The first system actually under discussion when Galileo published his book was a Tychonic geo-heliocentric system with diurnal rotation, Christen Longomontanus (1562–1647), Tycho’s chief assistant, had published an updated version based on Tycho’s data in his Astronomia Danica in 1622. This was the system that had been formally adopted by the Jesuits.

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The second was the elliptical heliocentric system of Johannes Kepler, of which I dealt with the relevant publications in the last post.

Galileo completely ignores Tycho, whose system could explain all of the available evidence for heliocentricity, because he didn’t want to admit that this was the case, arguing instead that the evidence must imply a heliocentric system. He also, against all the available empirical evidence, maintained his belief that comets were sublunar meteorological phenomena, because the supporters of a Tychonic system used their perceived solar orbit as an argument for their system.  He is even intensely disrespectful to Tycho in the Dialogo, for which Kepler severely castigated him. He also completely ignores Kepler, which is even more crass, as the best available arguments for heliocentricity were to be found clearly in Kepler published works. Galileo could not adopt Kepler’s system because it would mean that Kepler and not he would be the man, who proved the truth of the heliocentric system.

Although the first three days of the Dialogo provide a good polemic presentation for all of the evidence up till that point for a refutation of the Ptolemaic/Aristotelian system, with the very notable exception of the comets, Galileo’s book was out dated when it was written and had very little impact on the subsequent astronomical/cosmological debate in the seventeenth century. I will indulge in a little bit of hypothetical historical speculation here. If Galileo had actually written a balanced and neutral account of the positive and negative points of the Tychonic geo-heliocentric system with diurnal rotation and Kepler’s elliptical heliocentric system, it might have had the following consequences. Firstly, given his preeminent skills as a science communicator, his book would have been a valuable contribution to the ongoing debate and secondly he probably wouldn’t have been persecuted by the Catholic Church. However, one can’t turn back the clock and undo what has already been done.

I will close this overlong post with a few brief comments on the impact of the Church’s ban on the heliocentric theory, the heliocentric hypothesis was still permitted, and the trial and sentencing of Galileo, after all he was the most famous astronomer in Europe. Basically the impact was much more minimal than is usually implied in all the popular presentations of the subject. Outside of Italy these actions of the Church had almost no impact whatsoever, even in other Catholic countries. In fact a Latin edition of the Dialogo was published openly in Lyon in 1641, by the bookseller Jean-Antoine Huguetan (1567–1650), and dedicated to the French diplomat Balthasar de Monconys (1611–1665), who was educated by the Jesuits.

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Within Italy well-behaved Catholics censored their copies of Copernicus’ De revolutionibus according to the Church’s instructions but continued to read and use them. Censored copies of the book are virtually unknown outside of Italy. Also within Italy, astronomers would begin their discussions of heliocentricity by stating in the preface that the Holy Mother Church in its wisdom had declared this system to be false, but it is an interesting mathematical hypothesis and then go on in their books to discuss it fully. On the whole the Inquisition left them in peace.

 

***A brief footnote to the above: this is a historical sketch of what took place around 1630 in Northern Italy written from the viewpoint of the politics, laws and customs that ruled there at that time. It is not a moral judgement on the behaviour of either the Catholic Church or Galileo Galilei and I would be grateful if any commentators on this post would confine themselves to the contextual historical facts and not go off on wild moral polemics based on hindsight. Comments on and criticism of the historical context and/or content are, as always, welcome.

 

 

 

 

 

 

 

 

 

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Christmas Trilogy Part 3: The emergence of modern astronomy – a complex mosaic: Part XXVI

 

In popular presentations of the so-called scientific or astronomical revolutions Galileo Galilei is almost always presented as the great champion of heliocentricity in the first third of the seventeenth century. In fact, as we shall see, his contribution was considerably smaller than is usually claimed and mostly had a negative rather than a positive influence. The real champion of heliocentricity in this period was Johannes Kepler, who in the decade between 1617 and 1627 published four major works that laid the foundations for the eventual triumph of heliocentricity over its rivals. I have already dealt with one of these in the previous post in this series, the De cometis libelli tres I. astronomicus, theoremata continens de motu cometarum … II. physicus, continens physiologiam cometarum novam … III. astrologicus, de significationibus cometarum annorum 1607 et 1618 / autore Iohanne Keplero …, which was published in 1619 and as I’ve already said became the most important reference text on comets in the 1680’s during a period of high comet activity that we will deal with in a later post.

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Source: ETH Library Zurich

Chronologically the first of Kepler’s influential books from this decade was Volume I (books I–III) of his Epitome Astronomiae Copernicanae published in 1617, Volume II (book IV) followed in 1620 and Volume III (books V–VII) in 1621. This was a text book on heliocentric astronomy written in question and answer dialogue form between a teacher and a student spelling out the whole of heliocentric astronomy and cosmology in comparatively straight forward and simple terms, the first such textbook. There was a second edition containing all three volumes in 1635.

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Second edition 1635 Source

This book was highly influential in the decades following its publication and although it claims to be a digest of Copernican astronomy, it in fact presents Kepler’s own elliptical astronomy. For the first time his, now legendary, three laws of planetary motion are presented as such together. As we saw earlier the first two laws–I. The orbit of a planet is an ellipse and the Sun is at one of the focal points of that ellipse II: A line connecting the Sun and the planet sweeps out equal areas in equal times–were published in his Astronomia Nova in 1609. The third law was new first appearing in, what he considered to be his opus magnum, Ioannis Keppleri Harmonices mundi libri V (The Five Books of Johannes Kepler’s The Harmony of the World) published in 1619 and to which we now turn our attention.

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Source: Wikimedia Commons

Kepler’s first book was his Mysterium Cosmographicum published in 1597 with its, to our way of thinking, somewhat bizarre hypothesis that there are only six planets because the spaces between their orbits are defined by the five regular Platonic solids.

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Kepler’s Platonic solid model of the Solar System from Mysterium Cosmographicum Source: Wikimedia Commons

Although his calculation in 1597 showed a fairly good geometrical fit for his theory, it was to Kepler’s mind not good enough and this was his motivation for acquiring Tycho Brahe’s newly won more accurate data for the planetary orbits. He believed he could quite literally fine tune his model using the Pythagorean theory of the harmony of the spheres, that is that the ratio of the planetary orbits build a musical scale that is only discernable to the enlightened Pythagorean astronomer. The Harmonices Mundi was that fine tuning.

The first two books of the Harmonices Mundi layout Kepler’s geometrical theory of music, which geometrical constructions produced harmonious musical intervals and which disharmonious ones, based on which are constructible with straight edge and compass, harmonious, and which are not, disharmonious. The third book is Kepler’s contribution to the contemporary debate on the correct division of the intervals of the musical scale, in which Vincenzo Galilei (1520–1591), Galileo’s father, had played a leading role. The fourth book is the application of the whole to astrology and the fifth its application to astronomy and it is here that we find the third law.

In the fifth Kepler compare all possible ratios of planetary speeds and distances constructing musical scales for planets and musical intervals for the relationship between planets. It is here that he, one could say, stumbles upon his third law, which is known as the harmony law. Kepler was very much aware of the importance of his discovery as he tells us in his own words:

“After I had discovered true intervals of the orbits by ceaseless labour over a very long time and with the help of Brahe’s observations, finally the true proportion of the orbits showed itself to me. On the 8th of March of this year 1618, if exact information about the time is desired, it appeared in my head. But I was unlucky when I inserted it into the calculation, and rejected it as false. Finally, on May 15, it came again and with a new onset conquered the darkness of my mind, whereat there followed such an excellent agreement between my seventeen years of work at the Tychonic observations and my present deliberation that I at first believed that I had dreamed and assumed the sought for in the supporting proofs. But it is entirely certain and exact that the proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances.”

Translated into modern notation the third law is P12/P22=R13/R23, where P is the period of a planet and R is the mean radius of its orbit. It can be argues that this was Kepler’s greatest contribution to the history of the emergence of heliocentricity but rather strangely nobody really noticed its true significance until Newton came along at the end of the seventeenth century.

However they should have done because the third law gives us is a direct mathematical relationship between the size of the orbits of the planets and their duration, which only works in a heliocentric system. There is nothing comparable for either a full geocentric system or for a geo-heliocentric Tychonic or semi-Tychonic system. It should have hit the early seventeenth-century astronomical community like a bomb but it didn’t, which raises the question why it didn’t. The answer is because it is buried in an enormous pile of irrelevance in the Harmonices Mundi and when Kepler repeated it in the Epitome he gave it no real emphasis, so it remained relatively ignored.

On a side note, it is often thought that Kepler had abandoned his comparatively baroque Platonic solids concepts from the Mysterium Cosmographicum but now that he had, in his opinion, ratified it in the Harmonices Mundi he published a second edition of the book in 1621.

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Second Edition 1621 Source

Ironically the book of Kepler’s that really carried the day for heliocentricity against the geocentric and geo-heliocentric systems was his book of planetary tables based on Tycho Brahe’s data the Tabulae Rudolphinae (Rudolphine Tables) published in 1627, twenty-eight years after he first began working on them. Kepler had in fact been appointed directly by Rudolph II in Prague to produce these tables at the suggestion of Tycho in 1601. Turning Tycho’s vast collection of data into accurately calculated tables was a horrendous and tedious task and over the years Kepler complained often and bitterly about this burden.

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Tabulae Rudolphinae The frontispiece presents in graphic form a potted history of Western astronomy Source

However, he persevered and towards the end of the 1620s he was so far. Because he was the Imperial Mathematicus and had prepared the tables under the orders of the Emperor he tried to get the funds to cover the printing costs from the imperial treasury. This proved to be very difficult and after major struggles he managed to acquire 2000 florins of the more than 6000 that the Emperor owed him, enough to pay for the paper. He began printing in Linz but in the turmoil of the Thirty Years War the printing workshop got burnt down and he lost the already printed pages. Kepler decamped to Ulm, where with more difficulties he succeeded in finishing the first edition of 1000 copies. Although these were theoretically the property of the Emperor, Kepler took them to the Frankfurt book fair where he sold the entire edition to recoup his costs.

The Tabulae Rudolphinae were pretty much an instant hit. The principle function of astronomy since its beginnings in Babylon had always been to produce accurate tables and ephemerides for use initially by astrologers and then with time also cartographers, navigators etc. Astronomical systems and the astronomers, who created them, were judged on the quality and accuracy of their tables. Kepler’s Tabulae Rudolphinae based on Tycho’s data were of a level of accuracy previous unknown and thus immediately won many supporters. Those who used the tables assumed that their accuracies was due to Kepler’s elliptical planetary models leading to a gradually increasing acceptance of heliocentricity but this was Kepler’s system and not Copernicus’. Supported by the Epitome with the three laws of planetary motion Kepler’s version of heliocentricity became the dominant astronomical/cosmological system over the next decades but it would be another thirty to forty years, long after Kepler’s death, before it became the fully accepted system amongst astronomers.

 

 

 

 

 

 

 

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

One single occurrence in the year 1618 played a significant role in the history of astronomy, the appearance of a great comet. In fact there were three comets in 1618 but only one of them, a so-called great comet, made history. Great comets are those that are not just visible to astronomers but those that put on a spectacular display for the general public, the most well known being Comet Halley. The great comet of 1618 is important for the emergence of modern astronomy for three different reasons.

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An image of the great comet of 1618 as seen above the Southern German city of Augsburg

The first is that the Swiss, Jesuit Johann Baptist Cysat (c. 1587–1657), a student of Christoph Scheiner (1573–1650), who had assisted Scheiner with his sunspot observations,

Cysatus

Johann Baptist Cysat, holding a Jacob’s staff Source: Wikimedia Commons

became the first astronomer to observe a comet with a telescope giving the first ever description of a comet’s nucleus in his Mathemata astronomica de loco, motu, magnitudine et causis cometae qui sub finem anni 1618 et initium anni 1619 in coelo fulsit. Ingolstadt Ex Typographeo Ederiano 1619 (Ingolstadt, 1619). He followed Tycho Brahe in believing that comets orbited the sun. He also demonstrated the orbit was parabolic not circular.

Johannes Kepler visited Cysat in Ingolstadt and the two astronomers corresponded, however only one of their letters is extant. Kepler, naturally, also observed the great comet of 1618 and wrote and published one of his, often overlooked but actually important books, on it and comets in general in 1619, his De cometis libelli tres I. astronomicus, theoremata continens de motu cometarum … II. physicus, continens physiologiam cometarum novam … III. astrologicus, de significationibus cometarum annorum 1607 et 1618 / autore Iohanne Keplero …

This is one of the most important publications on comets in the seventeenth century and became a standard reference work in the 1680s, when a very major and very important debate was launched by the appearance of several bright comets. Unlike Cysat, Kepler thought that the trajectory of comets was rectilinear and only appeared curved because of the movement of the earth, he of course being a supporter of heliocentricity whereas Cysat supported geo-heliocentricity.

Kepler’s view on the trajectory of comets is interesting because already in 1610, having read Astronomia Nova, Sir William Lower (1570-1615), English landowner, member of parliament, and student and friend of Thomas Harriot, who together with Harriot had made careful observations of Comet Halley in 1607, suggested that comets also follow an elliptical orbit. However, his very perceptive views on the topic remained largely unknown at the time.

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Cartoon of Sir William Lower with telescope John Prydderch his neighbour and observing companion with a Jacob staff Artist: The initials “JMS” appear in the corner of the cartoon. The front cover of the Journal of the Astronomical Society of Wales in the late 1890’s, also hand drawn, bore this insginia and stood for J. M. Staniforth, the artist-in-chief of the Western Mail newspaper

Another astronomer who made careful observations of all three of the 1618 comets was the Jesuit Orazio Grassi (1583–1654).

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Orazio Grassi

Grassi published his views, which he had expounded as a lecture at the Collegio Romano, the Jesuit university, as De Tribus Cometis Anni MDCXVIII also in 1619.

Orazio_Grassi_De_tribus_cometis

Source: Wikimedia Commons

Like Cysat and Kepler he concluded that the comets were definitely supralunar and like Cysat, following Tycho, he concluded that they orbited the sun, like Tycho seeing this as a confirmation of the geo-heliocentric hypothesis. Grassi’s publication is notorious in the history of astronomy because it unintentionally launched one of the most infamous disputes in the history of astronomy, infamous because his opponent was none other than Galileo Galilei.

Galileo had not actually observed the comets of 1618 being confined to his bed by illness but was asked for his opinion on them by one of his fellow members of the Accademia dei Lincei, Virginio Cesarini (1595–1624).

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Virginio Cesarini, by Van Dyck Source: Wikimedia Commons

Galileo published his views, as a pernicious attack on Grassi’s De Tribus Cometis under the title Discorso delle Comete also in 1619 with a student of his, Mario Guiducci (1583–1646), as the named author, although it was an open secret that Galileo was the main author. Although Galileo always maintained that Guiducci and not he was the author, the manuscript that is still extant is largely in Galileo’s own hand and those paragraphs written by Guiducci have been corrected by Galileo.

Discorso_delle_comete

Source: Wikimedia Commons

Surprisingly in Discorso delle Comete Galileo turns against the, in the meantime, general consensus amongst astronomers formed over the best part of one hundred years of comet observations that comets are supralunar celestial phenomena and argues instead, with Aristotle, that they are sublunar meteorological phenomena. He argued that the failure to determine parallax was not because comets were distant objects but because they were simply optical phenomena that like rainbows display no parallax. This theory, of course, directly contradicts the telescopic observations of the comet made by Cysat. Galileo might not of known of Cysat’s observations or he might simply have ignored them. It should be remembered that in all his writings on heliocentricity, Galileo totally ignored the writings of Kepler, although he was well aware of them. What Galileo is actually arguing against is the correct view expressed by both Cysat and Grassi that comets orbit the sun, as they use this to bolster a Tychonic geo-heliocentric model of the cosmos, which is unacceptable to Galileo. Galileo also took the opportunity in the Discorso to take another swing at Scheiner on the subject of sunspots.

Under the pseudonym Lotario Sarsi Sigensanso, an anagram of his Latin name (his original lecture had been published anonymously, as was usual for Jesuits), Grassi published an answer to the Galileo/Guiducci Discorso delle Comete, his Libra astronomica ac philosophica also in 1619.

Grassi_-_Libra_astronomica_ac_philosophica,_1619_-_212632_F

Source: Wikimedia Commons

Here he presented scientific arguments and experimental tests to show that comets were definitely not optical phenomena. This of course further angered Galileo, who responded with one of his most famous publications Il Saggiatore, put out by the Accademia dei Lincei in 1623. Here we see a famous historical play on words. Grassi’s title Libra astronomica translates as The Astronomical Balance, i.e. weighing up the arguments, and Galileo’s answer Il Saggiatore translates as The Assayer.

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Source: Wikimedia Commons

One general point that has to be made in this dispute is that although Grassi fought his corner energetically, even going so far as to suggest that some of Galileo’s arguments were theologically suspect, his tone was always correct and courteous. Opposed to this, Galileo was constantly aggressive, abusive and insulting not only towards Grassi but also towards all the others such as Tycho and Scheiner, who appeared in his polemical crosshairs. This continued, in fact was even amplified, in Il Saggiatore, where he took on all those he considered to be his intellectual adversaries. As well as viciously attacking Grassi he openly accused both Simon Marius and Christoph Scheiner of plagiarism, false accusations in both cases.

Il Saggiatore is famous for its ‘Two Books’ passage and Galileo’s statement that ‘the book of nature is written in the language of mathematics’. One should point out that Galileo actually wrote geometry not mathematics, as it is usually translated. The two books topos was not new with Galileo but goes back to the Middle Ages. People get all excited by the language of mathematics quote but by the time Galileo wrote it in 1623 it was a commonplace. Copernicus’ De revolutionibus has the statement, “Let no one untrained in geometry enter here.” Copernicus very definitely wanted to take the explanation of the cosmos away from the philosophers and give it to the mathematicians. This sentiment was certainly shared amongst others by Gerolamo Cardano (1501–1576), Niccolò Fontana Tartaglia (1499–157), Giambattista Benedetti (1530–1590), who anticipated much of Galileo’s work on the laws of fall, Tycho Brahe (1546–1601), Christen Longomontanus (1562–1647), John Dee (1527–1608), Thomas Digges (c. 1546–1595), Thomas Harriot (c. 1560–1621), Johannes Kepler (1571–1630), Simon Stevin (1548­–1620) as well as Christoph Clavius (1538–1612) and all the excellent Jesuit mathematical scholars who came out of his programme of mathematical education at the Collegio Romano. When Galileo coined his phrase in 1623 he was rather stating the obvious and not something spectacularly new. The so-called scientific revolution has been explicitly referred to as the mathematization of science, a process that was well underway before Galileo dropped his platitude into the debate. This was even acknowledged by some of the prominent opponents to this development such as the occultists Giordano Bruno (1548–1600) and Robert Fludd (1574–1637), who both explicitly rejected it.

The other point to note, which gets over looked by most people, is that Galileo’s statement was made in the specific dispute that I have just sketched above. In this dispute it was Grassi, who argued scientifically and mathematically, whereas Galileo didn’t.

One thing that Galileo could do better than any of his rivals was write and Il Saggiatore is a masterpiece of invective and polemic, leading people to oversee or ignore the fact that Galileo was actually in the wrong. The text was immensely popular, which led to two developments. Firstly, because of his attacks on Grassi and Scheiner he alienated the Jesuits the most mathematically competent scholars, not only in the Catholic Church but also in the whole of Europe. It should be remembered that it was the Jesuits at the Collegio Romano, who had provided the very necessary confirmation of all of Galileo’s telescopic discoveries and also celebrated him for them. He would no longer be able to rely on their support in the future. Secondly it found favour with Maffeo Barberini (1568–1644), who in 1623 was elected Pope Urban VIII. Barberini was an old friend and supporter of Galileo’s and a close associate of the Accademia dei Lincei, who celebrated his election and amongst other things presented him with a copy of Il Saggiatore, which he enjoyed immensely. He granted Galileo a couple of private audiences, an unusual honour for a mere mathematicus, and Galileo became a Papal favourite, something that would prove to be a two edged sword.

 

 

 

 

 

 

 

 

 

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