Category Archives: Mediaeval Science

Renaissance Science – IV

We have now reached the period of history that the majority of people automatically think of when the hear or read the name, The Renaissance. The majority probably also think, when the hear the term, of a period in European art history, often called the Italian Renaissance, doing which the great artists Leonardo, Michelangelo, Raphael et al flourished. This is one aspect of the Renaissance that won’t be dealt with directly in this series but, of which some aspects do turn up on the fringes a couple of time. For a long time, the Renaissance was simply called the Renaissance, but because historians began to use the term for the other renaissances that we have already looked at–the Carolingian Renaissance, the Ottonian Renaissance and the Scientific Renaissance–it became common practice, at least amongst historians, to qualify the name as the Humanist Renaissance and it is here that we meet our first problem. Both the term Humanist and the term Renaissance were actually first coined in the nineteenth century. Somebody in the Early Modern period would not have recognised this name. So, what was it called then? It wasn’t. Although, as we will see the people, who kicked off the Renaissance distanced themselves from the Middle Ages, a term that they created, they gave their movement a name, but didn’t give their period one. Who were theses people, when were they active and what did they set out to do?

Before we examine the true origins of the Renaissance, we need to first dispel an oft repeated false statement. It is very common to read that the Renaissance started with the final collapse of the Eastern Roman Empire, when the Ottoman Turks captured Constantinople in 1453, with images of Greek scholars fleeing to Western Europe with bundles of Greek manuscript clutched under their arms. This is a myth. In fact, the Renaissance had its beginnings more than a hundred years earlier centred on Florence in Northern Italy.

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The final siege of Constantinople, contemporary 15th-century French miniature. Bertrandon de la Broquière in Voyages d’Outremer – http://www.bnf.fr Source: Wikimedia Commons

The earliest phase of the Renaissance is attributed to the writers Dante Alighieri (c. 1265–1321), Giovanni Boccaccio (1313–1375) and Francesco Petrarca (1304–1374), better known in English as Petrarch, who are considered to have launched a new wave of literature in the fourteenth century.

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Dante Alighieri, attributed to Giotto, in the chapel of the Bargello palace in Florence. This oldest picture of Dante was painted just prior to his exile and has since been heavily restored. Source: Wikimedia Commons

In its initial phase their Renaissance was a literary and linguistic movement. Led by Petrarch, the notary Coluccio Salutati (1331–1406) famous for his skills as a writer and orator, and the scholars Niccolò de’ Niccoli (1364–1437) and Poggio Bracciolini (1380–1459), this literary movement turned to classical Rome, as its model in literature and oratory.

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Petrarch portrait by Altichiero Source: Wikimedia Commons

In particular these men praised and tried to emulate the works of Marcus Tullius Cicero (106–43 BCE) and Marcus Fabius Quintilianus (c. 35–c. 100 CE), usually simple known as Cicero and Quintilian, both regarded as masters of oratory.

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First-century AD bust of Cicero in the Capitoline Museums, Rome via Wikimedia Commons

Their late medieval admirers regarded both the literary style and their classical Latin as exemplary and considered both style and language worthy of emulation. It is here that we witness the first rupture with the Middle Ages. The literary scholars of Northern Italy regarded the medieval Latin of the Church and universities as degenerate and barbaric and strove to replace it with, what they perceived to be, the pure uncorrupted classical Latin of Cicero. How successful they were can be seen in the fact that the Latin taught in schools and to archaeology and history undergraduates at universities in my youth in the 1970s was classical Latin and only classical Latin, medieval Latin still being regarded as somehow inferior, so that the medieval archaeologists and historians had to then subsequently learn medieval Latin. Of course, medieval Latin is not degenerate and corrupt, languages evolve and more than one thousand years separate Cicero and the twelfth century medieval university. Medieval Latin had evolved out of so-called Late Latin, the Latin that had developed between approximately the third and sixth centuries CE, influenced by both Christianity and the non-Latin languages spoken on the borders of the empire. Medieval Latin began to evolve around the seventh century heavily influenced by the Church and is also referred to as Ecclesiastical Latin. Compared to classical Latin, medieval Latin had a much larger vocabulary, because it needed terms not available in classical Latin, but also significant changes in grammar, syntax and orthography.

Having denigrated the medieval language those founders of the Renaissance, also dismissed the period itself, labelling it the Middle Ages, the period in-between the glory that was the classical period of Rome and their own almost as glorious revival of it. They didn’t actually label their own period but did refer to it in Italian, as rinascimento, a rebirth, which is of course the origin of the modern term Renaissance. They referred to their own activities as studia humanitatis, from the Latin humanitas meaning education befitting a cultivated man. Once again, the origin of the modern words: humanism, humanist, and the name, the humanities. These student of humanitas devoted themselves to searching out manuscripts in monastic libraries in Latin but also in Greek that fulfilled their concept of such an education, history, music, art, literature and poetry predominating. Poggio Bracciolini was particularly zealous finding many such manuscripts including Lucretius’ De rerum natura, Vitruvius’ De architectura and lost orations by Cicero and Quintilian.

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Frontispiece of a 1720 edition of the Institutio Oratoria, showing Quintilan teaching rhetoric Copper engraving by F. Bleyswyk. Source: Wikimedia Commons

These scholars also began to apply philological principles to the study of the manuscripts they recovered. The word itself is a fourteenth century coinage philologie meaning love of literature; personification of linguistics and literary knowledge. Aware that the oft copied manuscripts of ancient knowledge were corrupted by scribal errors and slips, they began to compare and analyse manuscripts, to discovery and irradicate those error and in so doing attempting to recreate the texts in their original state.

The initial impact of this movement on the medieval university was relatively small, although as we’ll see in later episode it did set other greater changes in motion. In this early phase the humanist scholars succeeded in reshaping the trivium removing logic so it was now grammar, rhetoric, history, moral philosophy and above all poetics. Impact of the latter can be clearly seen in later times. Georg von Peuerbach (1423–1461) a central figure in the history of astronomy, as a member of the First Viennese School of Mathematics, who was himself an accomplished poet, actually lectured on poetics at the university; his astronomy was, so to speak, an unofficial activity. Conrad Celtis (1459–1508), instrumental in introducing and spreading humanism north of the Alps and known in German as the Arch-Humanist, a crowned poet laureate and founder of the Second Viennese School of Mathematics, when called to the University of Vienna in 1497 founded a Collegium poetarum et mathematicorum, that is a college for poetry and mathematics.

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Conrad Celtis: Gedächtnisbild von Hans Burgkmair dem Älteren, 1507 Source: Wikimedia Commons

A question remains open, is it correct to name an entire epoch or period of history after what was initially a small, rather local movement within a limited academic sphere? The answer is yes, because that movement created waves that spread through time and space outwards from Florence to encompass the whole of Europe and influence the intellectual and academic development over the next two hundred plus years. In later posts we shall be looking at those developments with regard to their impact on the evolution of the sciences. Another open question is when did the Renaissance end? This is hotly debated, and I shall, for my purposes, follow Francis Yates, who takes the end of the Thirty Years War as the end of the Renaissance, which I will explain, or justify in my next post. A closing important comment is that there is actually a very high level of continuity rather than disruption from the High Middle Ages through the Renaissance and one can regard the Renaissance both as a phase of the Middle Ages but also of the Early Modern Period; all historical periodisations are of course artificial and also to some extent arbitrary.

 

 

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Renaissance Science – III

Before we can finally move on to the actually subject of this series, The Renaissance, we first need to take a brief look at the European medieval university to which the Renaissance was to some extent a reaction. Actually, the European is superfluous as the medieval university is a unique European invention but it has become a bad habit in recent years to label different institutions of higher education from other cultures universities, particularly when claiming that they are older. Yes, other culture had institutes of higher education, many of them much earlier than the medieval universities. For example, the ancient Greek schools of philosophy were institutions of higher education. But the institutions of higher education of each culture have different roots, different structures and different aims and labelling them all universities is simply wrong. A madrasa is not a university and a university is not a madrasa. To use the term university exclusively for the medieval European institution also does not imply some sort of superiority, which some people try to suggest is what is wrong with this exclusive usage. 

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View of the Qarawiyyin Mosque on the skyline of central Fes el-Bali: the green-tiled roofs of the prayer hall and the minaret (white tower on the left) are visible.Founded by Fatima al-Fihri in 859 and falsely called the oldest university in the world Source: Wikimedia Commons

The European universities have their roots in the cathedral schools that began to appear in the seventh century, following the collapse of the Western Roman Empire. The education in these institutions was nominally based on the seven liberal arts, an educational ideal that goes back to the Pythagoreans. It consists of the trivium–grammar, logic and rhetoric–and the quadrivium–arithmetic, geometry, music and astronomy. However, if we look at the description of the quadrivium by Isidore of Seville (c. 560­–636), in his Etymologiae, offered at the early cathedral schools, arithmetic, geometry and music are little more than a short list of empty definition with only astronomy having some substance as a discipline. This schools taught little more than Latin and the basics of Christian theology. 

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A page of Etymologiae, Carolingian manuscript (8th century), Source: Brussels, Royal Library of Belgium via Wikimedia Commons

The Carolingian Renaissance, which I described in the first post of this series was basically an upgrading of the cathedral schools to proper institutes of education with a fairly low level but much fuller curriculum. Between the tenth and twelfth centuries a series of great teaching masters very much raised the standard of cathedral schools and increased and improved the scientific content of the curriculum. These men attracted large numbers of students and trained other teaching masters. Amongst the most well-known were Gerbert of Arillac (c. 964–1003), Adalberon of Laon (d. 1030), John of Auxerre, Thierry of Chartres (d. c. 1150), Fulbert of Chartres (c.960–1128) a pupil of Gerbert, Peter Abelard (c. 1079–1142), Bernard of Chartres (d. after 1124), William of Conches (c. 1090–c. 1160) pupil of Bernard of Chartres, Clarenbold of Arras (c. 1110–c. 1187) also school of Chartres, and John of Salisbury (late 1110s–1180), a pupil of William of Conches. Most of these were Neo-Platonists heavily influenced by the Timaeus, one of the few Greek natural philosophy texts known throughout the Middle ages.

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Chartres Cathedral by night Source: Wikimedia Commons

Europe saw some major changes in the period between 800 CE and 1200 CE, which was a comparatively stable political period. The major changes were in agriculture. In this period the most important innovation was the mouldboard plough and the related heavy plough. Along with this was the invention of the horse collar and the horseshoe, which meant that the horse could replace the ox as the ploughing animal. This meant that much heavier land could be used for crop production and ploughing took much less time. Another significant improvement was the introduction of a three-field rotation system to replace the earlier two-field one. This led to a major increase in food production, which in turn led to a large population increase.

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Medieval horse drawn heavy plough Source: Wikimedia Commons

This was paralleled by a growth in the town and cities with more and more people moving from rural to urban residency. The same period saw a major economic upturn within Europe with a substantial increase in long distance trading and a move to a money-based economy.  

These developments led to the transition of some of the cathedral schools to becoming the first universities. The towns and cities attracted increasing numbers of students looking for teachers and teachers looking for students. A major change came in the twelfth century with the rediscovery of the Corpus Juris Civilis. The codification of Roman law was originally created in the sixth century but almost totally disappeared during the Early Middle Ages. This reintroduced the concept of a corporation from the Latin corpus meaning body or body of people recognised as a legal entity. This led to merchant traders and artisans forming corporations, known in the case of artisans as guilds, with legally defined membership, giving the members a collective legal status and collective protection under the law.

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German medieval guild symbols

The travelling students and masters, as individuals living in cities other than their hometowns and cities, had very little legal status or legal protection, so they too formed corporations, for which one Latin term was universitas, meaning whole or the whole. There were universitas magistrorum or universities of masters, universitas scholarium or universities of students and universitas magistrorum et scholarium or universities of masters and students.

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universitas scholarium

Originally universitas referred to these corporate bodies and also to individual faculties, such as the faculty of arts, faculty of theology, faculty of law or faculty of medicine. The masters and students of each faculty forming their own corporation or universitas. The term for what we now call a university was studium generale, which was only applied to a school with at least three of the four traditional faculties or was a highly prestigious school such as Paris, Oxford and Bologna, or both. 

At some point the term universitas ceased to be used for corporations of traders and crafts guilds being then only used for academic corporation, as a consequence universitas began to replace studium generale for the whole academic institution. The big three–Paris, Oxford and Bologna–were the first to become universities in something approaching the modern meaning of the term. As there was a gradual transition from cathedral school to university it is impossible to say exactly when any of them became a university, but it is general acknowledged that in each case it occurred before twelve hundred with Bologna the first medieval university. Bologna concentrated more on law and theology, whereas Paris and Oxford concentrated more on philosophy. By fifteen hundred there were about seventy European university with, in general, those in Northern Europe following the Paris-Oxford model and those in the South and Italy modelled on Bologna. 

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Bologna University Interior view of the Porticum and Loggia of its oldest College, the Royal Spanish College. Source: Wikimedia Commons

A full university had four faculties, arts, law, medicine and theology. The arts faculty was the undergraduate faculty where nominally the seven liberal arts determined the curriculum. Here the first degree, BA, usually took four years but many students left the university after only two years, without a degree, having acquired the basic minimum of an education. Those who stayed after completing a BA, went on to acquire and MA, which was the teaching qualification and usually required a further two years of study. It was these MAs, who taught the undergraduates. Those with a MA could now progress to one of the higher faculties, law, medicine and theology, progressing through BA and MA till they finally graduated with a doctorate. This course of studies took a substantial number of years, so the number of students, who followed this course always remained relatively small. 

It is no coincidence that the emergence of the universities coincided with the highpoint of the translation movement or Scientific Renaissance, and the texts brought into the European sphere had a major influence on the curriculum of the new universities. 

The newly acquired knowledge radically upgraded the quadrivium with the first six books of Euclid’s Elements becoming the geometry course, arithmetic remained anchored in Boethius’ De institutione arithmetica, which was largely a Latin translation of the Introduction to Arithmetic of the Neopythagorean Nicomachus of Gerasa (c. 60–c. 120 CE). This was complimented by the study of Algorismus, that is the Hindu-Arabic number system, used in computos, the calculation of the date of Easter.  Music was also taken from Boethius’ De musica in turn based on a lost work of Nicomachus and Ptolemaeus’ Harmonica. Over the course of the next three centuries the works of Boethius were replaced by new texts written by medieval masters. Astronomy was largely taught according to John of Sacrobosco’s (c. 1195–c. 1256) Tractatus de Sphera (c. 1230). Sacrobosco taught at the university of Paris and also wrote a widely used Algorismus, De Arte Numerandi. Because Sacrobosco’s Sphera was very basic it was complimented with a Theorica planetarum, by an unknown medieval author, which dealt with elementary planetary theory and a basic introduction to the cosmos.

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13th century Manuscript of Sacrobosco’s Tractatus de Sphera, manuscript also contains his Algorimus Source

It should, however, be noted that the commitment to actually teaching the quadrivium during the High Middle Ages was in practice very low at most medieval universities. Lectures on the quadrivium were often only held on holidays, when normal teaching was suspended. The quadrium subjects were normally not examination subjects and it was even the case at many universities that if a student did not have the credit for a quadrivium course, he could acquire it simply by paying the lecture fees.

The biggest change, however, was in the trivium, which became basically the works of Aristotle. Having acquired a fairly complete model of the world and everything in it, in the works of Aristotle, the medieval scholars adopted it. This meant that the natural philosophy taught at the universities consisted mainly of Aristotle’s physics, meaning the general study of nature, and his cosmology. This was not necessarily that simple, as the universities were institutions of the Church and Aristotle was a pagan and various aspects of his philosophy contradicted the Church’s teachings. The biggest stumbling block was that Aristotle believed in an eternal cosmos with no beginning, whereas a central tenet of Christianity was the creation of the world by God, as described in Genesis. There were other philosophical problems that we don’t need to analyse in detail here. 

Given the conflicts there were various attempts by powerful figures in the Church, particularly in Paris, to try to ban the study of Aristotle in the universities. The most famous one being the list of 219 philosophical and theological propositions issued by the Bishop of Paris, Étienne Tempier (d. 1279) in 1277, which were contradictory to Christian belief and should not be taught at the university.

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Bishop Étienne Tempier

All such prohibitions failed to have a real affect but did create an interesting new method of thought into the medieval discourse. Scholars began to discuss these banned theses hypothetically, i.e., what if the universe were eternal or what if the Earth rotated on its axis once a day and the sphere of the fixed stars were still. One important point is that medieval Aristotelean philosophy was not Aristotle’s philosophy. Things had changed and progressed over the centuries; the most well-known example is that the impetus theory had replaced Aristotle’s theory of projectile motion. Also, thought was not as static on the medieval university, as it is often described, especially by the humanist scholars rebelling against the Aristotelean tradition in the Renaissance.

In the higher faculties it is only the faculty of medicine that is of direct interest for the history of science, although as we saw above the theologians determined what was permitted and what not. The curriculum of the faculty of medicine was informed with translations from Greek physicians, mostly Galen and Hippocrates but the major influence was Arabic medical texts, which were also based on the works of Galen and Hippocrates. One of the biggest were The Canon of Medicine (Al-Qanun fi’t-Tibb) a five-volume medical encyclopaedia written by Ibn Sina, known in the Middle Ages as Avicenna, which remained a central European university text for several centuries.

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The Canon of Medicine (Al-Qanun fi’t-Tibb) Ibn Sina, known in the Middle Ages as Avicenna

Another was Kitāb al-Ḥāwī fī al-ṭibb (The Comprehensive Book on Medicine) by Abū Bakr Muhammad Zakariyyā Rāzī, known in the Middle Ages as Rhazes. Another nine-volume medical encyclopaedia. There were also many other Arabic texts translated into Latin. This predominance of Arabic influence would come to play a role in the changes demanded during the Renaissance.

It is important to note that medieval university knowledge, even in medicine, was literary or book knowledge, that is totally theoretical without any practical aspects. Scholars challenged the ideas of other scholars with theoretical arguments not with experiments or newly acquired empirical evidence. As we shall see this is the basis for the major change that took place during the Renaissance.

The above is, of course, a simplified sketch of a process that should have a complete series of its own but I hope will suffice as a background to the changes that took place during the Renaissance the actual subject of this series.

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Renaissance Science – I

To paraphrase what is possibly the most infamous opening sentence in a history of science book[1], there was no such thing as Renaissance science, and this is the is the start of a blog post series about it. Put another way there are all sorts of problems with the term or concept Renaissance Science, several of which should entail abandoning the use of the term and in a later post I will attempt to sketch the problems that exist with the term Renaissance itself and whether there is such a thing as Renaissance science? Nevertheless, I intend to write a blog post series about Renaissance science starting today.

We could and should of course start with the question, which Renaissance? When they hear the term Renaissance, most non-historians tend to think of what is often referred to as the Humanist Renaissance, but historians now use the term for a whole series of period in European history or even for historical periods in other cultures outside of Europe.

Renaissance means rebirth and is generally used to refer to the rediscovery or re-emergence of the predominantly Greek, intellectual culture of antiquity following a period when it didn’t entirely disappear in Europe but was definitely on the backburner for several centuries following the decline and collapse of the Western Roman Empire. The first point to note is that this predominantly Greek, intellectual culture didn’t disappear in the Eastern Roman Empire centred round its capitol Constantinople. An empire that later became known as the Byzantine Empire. The standard myth is that the Humanist Renaissance began with the fall of Byzantium to the Muslims in 1453 but it is just that, a myth.

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Raphael’s ‘School of Athens’ (1509–1511) symbolises the recovery of Greek knowledge in the Renaissance Source: Wikimedia Commons

As soon as one mentions the Muslims, one is confronted with a much earlier rebirth of predominantly Greek, intellectual culture, when the, then comparatively young, Islamic Empire began to revive and adopt it in the eight century CE through a massive translation movement of original Greek works covering almost every subject. Writing in Arabic, Arab, Persian, Jewish and other scholars, actively translated the complete spectrum of Greek science into Arabic, analysed it, commented on it, and expanded and developed it, over a period of at least eight centuries.  It is also important to note that the Islamic scholars also collected and translated works from China and India, passing much of the last on to Europe together with the Greek works later during the European renaissances.

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The city of Baghdad 150–300 AH (767 and 912 CE) centre of the Islamic recovery and revival of Greek scientific culture Source: Wikimedia Commons

Note the plural at the end of the sentence. Many historians recognise three renaissances during the European Middle Ages. The first of these is the Carolingian Renaissance, which dates to the eighth and ninth century CE and the reigns of Karl der Große (742–814) (known as Charlemagne in English) and Louis the Pious (778–840).

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Charlemagne (left) and Pepin the Hunchback (10th-century copy of 9th-century original) Source: Wikimedia Commons

This largely consisted of the setting up of an education system for the clergy throughout Europe and increasing the spread of Latin as the language of learning. Basically, not scientific it had, however, an element of the mathematical sciences, some mathematics, computus (calendrical calculations to determine the date of Easter), astrology and simple astronomy due to the presence of Alcuin of York (c. 735–804) as the leading scholar at Karl’s court in Aachen.

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Rabanus Maurus Magnentius (left) another important teacher in the Carolignian Renaissance with Alcuin (middle) presenting his work to Otgar Archbishop of Mainz a supporter of Louis the Pious Source: Wikimedia Commons

Through Alcuin the mathematical work of the Venerable Bede (c. 673–735), (who wrote extensively on mathematical topics and who was also the teacher of Alcuin’s teacher, Ecgbert, Archbishop of York) flowed onto the European continent and became widely disseminated.

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The Venerable Bede writing the Ecclesiastical History of the English People, from a codex at Engelberg Abbey in Switzerland. Source: Wikimedia Commons

Karl’s Court had trade and diplomatic relations with the Islamic Empire and there was almost certainly some mathematical influence there in the astrology and astronomy practiced in the Carolingian Empire. It should also be noted that Alcuin and associates didn’t start from scratch as some knowledge of the scholars from late antiquity, such as Boethius (477–524), Macrobius (fl. c. 400), Martianus Capella (fl. c. 410–420) and Isidore of Seville (c. 560–636) had survived. For example, Bede quotes from Isidore’s encyclopaedia the Etymologiae.

The second medieval renaissance was the Ottonian Renaissance in the eleventh century CE during the reigns of Otto I (912–973), Otto II (955–983), and Otto III (980–1002). The start of the Ottonian Renaissance is usually dated to Otto I’s second marriage to Adelheid of Burgundy (931–999), the widowed Queen of Italy in 951, uniting the thrones of Germany (East Francia) and Italy, which led to Otto being crowned Holy Roman Emperor by the Pope in 962.

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Statues of Otto I, right, and Adelaide in Meissen Cathedral. Otto and Adelaide were married after his annexation of Italy. Source: Wikimedia Commons

This renaissance was largely confined to the Imperial court and monasteries and cathedral schools. The major influences came from closer contacts with Byzantium with an emphasis on art and architecture.

There was, however, a strong mathematical influence brought about through Otto’s patronage of Gerbert of Aurillac (c. 946–1003). A patronage that would eventually lead to Gerbert becoming Pope Sylvester II.

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Sylvester, in blue, as depicted in the Evangelistary of Otto III Source: Wikimedia Commons

A monk in the Monastery of St. Gerald of Aurillac, Gerbert was taken by Count Borrell II of Barcelona to Spain, where he came into direct contact with Islamic culture and studied and learnt some astronomy and mathematics from the available Arabic sources. In 969, Borrell II took Gerbert with him to Rome, where he met both Otto I and Pope John XIII, the latter persuaded Otto to employ Gerbert as tutor for his son the future Otto II. Later Gerbert would exercise the same function for Otto II’s son the future Otto III. The close connection with the Imperial family promoted Gerbert’s ecclesiastical career and led to him eventually being appointed pope but more importantly in our context it promoted his career as an educator.

Gerbert taught the whole of the seven liberal arts, as handed down by Boethius but placed special emphasis on teaching the quadrivium–arithmetic, geometry, music and astronomy–bringing in the knowledge that he had acquired from Arabic sources during his years in Spain. He was responsible for reintroducing the armillary sphere and the abacus into Europe and was one of the first to use Hindu-Arabic numerals, although his usage of them had little effect. He is also reported to have used sighting tubes to aid naked-eye astronomical observations.

Gerbert was not a practicing scientist but rather a teacher who wrote a series of textbook on the then mathematical sciences: Libellus de numerorum divisione, De geometria, Regula de abaco computi, Liber abaci, and Libellus de rationali et ratione uti.

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12th century copy of De geometria Source: Wikimedia Commons

His own influence through his manuscripts and his letters was fairly substantial and this was extended by various of his colleagues and students. Abbo of Fleury (c. 945–1004), a colleague, wrote extensively on computus and astronomy, Fulbert of Chartres (c. 960–1028), a direct student, also introduced the use of the Hindu-Arabic numerals. Hermann of Reichenau (1013–1054 continued the tradition writing on the astrolabe, mathematics and astronomy.

Gerbert and his low level, partial reintroduction into Europe of the mathematical science from out of the Islamic cultural sphere can be viewed as a precursor to the third medieval renaissance the so-called Scientific Renaissance with began a century later at the beginning of the twelfth century. This was the mass translation of scientific works, across a wide spectrum, from Arabic into Latin by European scholars, who had become aware of their own relative ignorance compared to their Islamic neighbours and travelled to the border areas between Europe and the Islamic cultural sphere of influence in Southern Italy and Spain. Some of them even travelling in Islamic lands. This Scientific Renaissance took place over a couple of centuries and was concurrent with the founding of the European universities and played a major role in the later Humanist Renaissance to which it was viewed by the humanists as a counterpart. We shall look at it in some detail in the next post.

[1] For any readers, who might not already know, the original quote is, “There was no such thing as the Scientific Revolution, and this is a book about it”, which is the opening sentence of Stevin Shapin’s The Scientific Revolution, The University of Chicago Press, Chicago and London, 1996

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Illuminating medieval science

 

There is a widespread popular vision of the Middle ages, as some sort of black hole of filth, disease, ignorance, brutality, witchcraft and blind devotion to religion. This fairly-tale version of history is actively propagated by authors of popular medieval novels, the film industry and television, it sells well. Within this fantasy the term medieval science is simply an oxymoron, a contradiction in itself, how could there possible be science in a culture of illiterate, dung smeared peasants, fanatical prelates waiting for the apocalypse and haggard, devil worshipping crones muttering curses to their black cats?

Whilst the picture I have just drawn is a deliberate caricature this negative view of the Middle Ages and medieval science is unfortunately not confined to the entertainment industry. We have the following quote from Israeli historian Yuval Harari from his bestselling Sapiens: A Brief History of Humankind (2014), which I demolished in an earlier post.

In 1500, few cities had more than 100,000 inhabitants. Most buildings were constructed of mud, wood and straw; a three-story building was a skyscraper. The streets were rutted dirt tracks, dusty in summer and muddy in winter, plied by pedestrians, horses, goats, chickens and a few carts. The most common urban noises were human and animal voices, along with the occasional hammer and saw. At sunset, the cityscape went black, with only an occasional candle or torch flickering in the gloom.

On medieval science we have the even more ignorant point of view from American polymath and TV star Carl Sagan from his mega selling television series Cosmos, who to quote the Cambridge History of Medieval Science:

In his 1980 book by the same name, a timeline of astronomy from Greek antiquity to the present left between the fifth and the late fifteenth centuries a familiar thousand-year blank labelled as a “poignant lost opportunity for mankind.” 

Of course, the very existence of the Cambridge History of Medieval Science puts a lie to Sagan’s poignant lost opportunity, as do a whole library full of monographs and articles by such eminent historians of science as Edward Grant, John Murdoch, Michael Shank, David Lindberg, Alistair Crombie and many others.

However, these historians write mainly for academics and not for the general public, what is needed is books on medieval science written specifically for the educated layman; there are already a few such books on the market, and they have now been joined by Seb Falk’s truly excellent The Light Ages: The Surprising Story of Medieval Science.[1]  

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How does one go about writing a semi-popular history of medieval science? Falk does so by telling the life story of John of Westwyk an obscure fourteenth century Benedictine monk from Hertfordshire, who was an astronomer and instrument maker. However, John of Westwyk really is obscure and we have very few details of his life, so how does Falk tell his life story. The clue, and this is Falk’s masterstroke, is context. We get an elaborate, detailed account of the context and circumstances of John’s life and thereby a very broad introduction to all aspects of fourteenth century European life and its science.

We follow John from the agricultural village of Westwyk to the Abbey of St Albans, where he spent the early part of his life as a monk. We accompany some of his fellow monks to study at the University of Oxford, whether John studied with them is not known.

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Gloucester College was the Benedictine College at Oxford where the monks of St Albans studied

We trudge all the way up to Tynemouth on the wild North Sea coast of Northumbria, the site of daughter cell of the great St Alban’s Abbey, main seat of Benedictines in England. We follow John when he takes up the cross and goes on a crusade. Throughout all of his wanderings we meet up with the science of the period, John himself was an astronomer and instrument maker.

Falk is a great narrator and his descriptive passages, whilst historically accurate and correct,[2] read like a well written novel pulling the reader along through the world of the fourteenth century. However, Falk is also a teacher and when he introduces a new scientific instrument or set of astronomical tables, he doesn’t just simply describe them, he teachers the reader in detail how to construct, read, use them. His great skill is just at the point when you think your brain is going to bail out, through mathematical overload, he changes back to a wonderfully lyrical description of a landscape or a building. The balance between the two aspects of the book is as near perfect as possible. It entertains, informs and educates in equal measures on a very high level.

Along the way we learn about medieval astronomy, astrology, mathematics, medicine, cartography, time keeping, instrument making and more. The book is particularly rich on the time keeping and the instruments, as the Abbott of St Albans during John’s time was Richard of Wallingford one of England’s great medieval scientists, who was responsible for the design and construction of one of the greatest medieval church clocks and with his Albion (the all in one) one of the most sophisticated astronomical instruments of all time. Falk’ introduction to and description of both in first class.

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The book is elegantly present with an attractive typeface and is well illustrated with grey in grey prints and a selection of colour ones. There are extensive, informative endnotes and a good index. If somebody reads this book as an introduction to medieval science there is a strong chance that their next question will be, what do I read next. Falk gives a detailed answer to this question. There is an extensive section at the end of the book entitled Further Reading, which gives a section by section detailed annotated reading list for each aspect of the book.

Seb Falk has written a brilliant introduction to the history of medieval science. This book is an instant classic and future generations of schoolkids, students and interested laypeople when talking about medieval science will simply refer to the Falk as a standard introduction to the topic. If you are interested in the history of medieval science or the history of science in general, acquire a copy of Seb Falk’s masterpiece, I guarantee you won’t regret it.

[1] American edition: Seb Falk, The Light Ages: The Surprising Story of Medieval Science, W. W. Norton & Co., New York % London, 2020

British Edition: Seb Falk, The Light Ages: A Medieval Journey of Discover, Allen Lane, London, 2020

[2] Disclosure: I had the pleasure and privilege of reading the whole first draft of the book in manuscript to check it for errors, that is historical errors not grammatical or orthographical ones, although I did point those out when I stumbled over them.

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

Our medieval technological inheritance.

“Positively medieval” has become a universal put down for everything considered backward, ignorant, dirty, primitive, bigoted, intolerant or just simply stupid in our times. This is based on a false historical perspective that paints the Middle Ages as all of these things and worse. This image of the Middle Ages has its roots in the Renaissance, when Renaissance scholars saw themselves as the heirs of all that was good, noble and splendid in antiquity and the period between the fall of the Roman Empire and their own times as a sort of unspeakable black pit of ignorance and iniquity. Unfortunately, this completely false picture of the Middle Ages has been extensively propagated in popular literature, film and television.

Particularly in the film and television branch, a film or series set in the Middle Ages immediately calls for unwashed peasants herding their even filthier swine through the mire in a village consisting of thatch roofed wooden hovels, in order to create the ‘correct medieval atmosphere’. Add a couple of overweight, ignorant, debauching clerics and a pox marked whore and you have your genuine medieval ambient. You can’t expect to see anything vaguely related to science or technology in such presentations.

Academic medieval historians and historians of science and technology have been fighting an uphill battle against these popular images for many decades now but their efforts rarely reach the general lay public against the flow of the latest bestselling medieval bodice rippers or TV medieval murder mystery. What is needed, is as many semi-popular books on the various aspects of medieval history as possible. Whereby with semi-popular I mean, written for the general lay reader but with its historical facts correct. One such new volume is John Farrell’s The Clock and the Camshaft: And Other Medieval Inventions We Still Can’t Live Without.[1]

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Farrell’s book is a stimulating excursion through the history of technological developments and innovation in the High Middle Ages that played a significant role in shaping the modern world.  Some of those technologies are genuine medieval discoveries and developments, whilst others are ones that either survived or where reintroduced from antiquity. Some even coming from outside of Europe. In each case Farrell describes in careful detail the origins of the technology in question and if known the process of transition into European medieval culture.

The book opens with agricultural innovations, the deep plough, the horse collar and horse shoes, which made it possible to use horses as draught animals instead of or along side oxen, and new crop rotation systems. Farrell explains why they became necessary and how they increased food production leading indirectly to population growth.

Next up we have that most important of commodities power and the transition from the hand milling of grain to the introduction of first watermills and then windmills into medieval culture. Here Farrell points out that our current knowledge would suggest that the more complex vertical water mill preceded the simpler horizontal water mill putting a lie to the common precept that simple technology always precedes more complex technology. At various points Farrell also addresses the question as to whether technological change drives social and culture change or the latter the former.

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Having introduced the power generators, we now have the technological innovations necessary to adapt the raw power to various industrial tasks, the crank and the camshaft. This is fascinating history and the range of uses to which mills were then adapted using these two ingenious but comparatively simple power take offs was very extensive and enriching for medieval society. One of those, in this case an innovation from outside of Europe, was the paper mill for the production of that no longer to imagine our society without, paper. This would of course in turn lead to that truly society-changing technology, the printed book at the end of the Middle Ages.

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Along side paper perhaps the greatest medieval innovation was the mechanical clock. At first just a thing of wonder in the towers of some of Europe’s most striking clerical buildings the mechanical clock with its ability to regulate the hours of the day in a way that no other time keeper had up till then gradually came to change the basic rhythms of human society.

Talking of spectacular clerical buildings the Middle Ages are of course the age of the great European cathedrals. Roman architecture was block buildings with thick, massive stonewalls, very few windows and domed roofs. The art of building in stone was one of the things that virtually disappeared in the Early Middle Ages in Europe. It came back initially in an extended phase of castle building. Inspired by the return of the stonemason, medieval, European, Christian society began the era of building their massive monuments to their God, the medieval cathedrals. Introducing architectural innovation like the pointed arch, the flying buttress and the rib vaulted roof they build large, open buildings flooded with light that soared up to the heavens in honour of their God. Buildings that are still a source of wonder today.

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In this context it is important to note that Farrell clearly explicates the role played by the Catholic Church in the medieval technological innovations, both the good and the bad. Viewed with hindsight the cathedrals can be definitely booked for the good but the bad? During the period when the watermills were introduced into Europe and they replaced the small hand mills that the people had previously used to produce their flour, local Church authorities gained control of the mills, a community could only afford one mill, and forced the people to bring their grain to the Church’s mill at a price of course. Then even went to the extent of banning the use of hand mills.

People often talk of the Renaissance and mean a period of time from the middle of the fifteenth century to about the beginning of the seventeenth century. However, for historians of science there was a much earlier Renaissance when scholars travelled to the boundaries between Christian Europe and the Islamic Empire in the twelfth and thirteenth centuries in order to reclaim the knowledge that the Muslims had translated, embellished and extended in the eight and ninth centuries from Greek sources. This knowledge enriched medieval science and technology in many areas, a fact that justifies its acquisition here in a book on technology.

Another great medieval invention that still plays a major role in our society, alongside the introduction of paper and the mechanical clock are spectacles and any account of medieval technological invention must include their emergence in the late thirteenth century. Spectacles are something that initially emerged from Christian culture, from the scriptoria of the monasteries but spread fairly rapidly throughout medieval society. The invention of eyeglasses would eventually lead to the invention of the telescope and microscope in the early seventeenth century.

Another abstract change, like the translation movement during that first scientific Renaissance, was the creation of the legal concept of the corporation. This innovation led to the emergence of the medieval universities, corporations of students and/or their teachers. There is a direct line connecting the universities that the Church set up in some of the European town in the High Middle Ages to the modern universities throughout the world. This was a medieval innovation that truly helped to shape our modern world.

Farrell’s final chapter in titled The Inventions of Discovery and deals both with the medieval innovations in shipbuilding and the technology of the scientific instruments, such as astrolabe and magnetic compass that made it possible for Europeans to venture out onto the world’s oceans as the Middle Ages came to a close. For many people Columbus’ voyage to the Americas in 1492 represents the beginning of the modern era but as Farrell reminds us all of the technology that made his voyage possible was medieval.

All of the above is a mere sketch of the topics covered by Farrell in his excellent book, which manages to pack an incredible amount of fascinating information into what is a fairly slim volume. Farrell has a light touch and leads his reader on a voyage of discovery through the captivating world of medieval technology. The book is beautifully illustrated by especially commissioned black and white line drawing by Ryan Birmingham. There are endnotes simply listing the sources of the material in main text and an extensive bibliography of those sources. The book also has, what I hope, is a comprehensive index.[2]

Farrell’s book is a good, readable guide to the world of medieval technology aimed at the lay reader but could also be read with profit by scholars of the histories of science and technology and as an ebook or a paperback is easily affordable for those with a small book buying budget.

So remember, next time you settle down with the latest medieval pot boiler with its cast of filthy peasants, debauched clerics and pox marked whores that the paper that it’s printed on and the reading glasses you are wearing both emerged in Europe in the Middle Ages.

[1] John W. Farrell, The Clock and the Camshaft: And Other Medieval Inventions We Still Can’t Live Without, Prometheus Books, 2020.

[2] Disclosure: I was heavily involved in the production of this book, as a research assistant, although I had nothing to do with either the conception or the actual writing of the book that is all entirely John Farrell’s own work. However, I did compile the index and I truly hope it will prove useful to the readers.

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Filed under Book Reviews, History of science, History of Technology, Mediaeval Science

Mathematics and natural philosophy: Robert G socks it to GG

In my recent demolition of Mario Livio’s very pretentious Galileo and the Science Deniers I very strongly criticised Livio’s repeated claims, based on Galileo’s notorious Il Saggiatore quote on the two books, that Galileo was somehow revolutionary in introducing mathematics into the study of science. I pointed out that by the time Galileo wrote his book this had actually been normal practice for a long time and far from being revolutionary the quote was actually a common place.

Last night whilst reading my current bedtime volume, A Mark Smith’s excellent From Sight to Light: The Passage from Ancient to Modern Optics,(University of Chicago Press, 2015) I came across a wonderfully appropriate quote on the topic from Robert Grosseteste (c.1175–1253). For those that don’t know Grosseteste was an English cleric who taught at Oxford University and who became Bishop of Lincoln. He played an important and highly influential role in medieval science, particularly in helping to establish optics as a central subject in the medieval university curriculum.

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An early 14th-century portrait of Grosseteste Source: Wikimedia Commons

Of course, this is problematic for Livio, who firmly labelled the Catholic Church as anti-science and who doesn’t think there was any medieval science, remember that wonderfully wrong quote:

Galileo introduced the revolutionary departure from the medieval, ludicrous notion that everything worth knowing was already known.

If this were true then medieval science would be an oxymoron but unfortunately for Livio’s historical phantasy there was medieval science and Grosseteste was one of its major figure. If you want to know more about Grosseteste then I recommend the Ordered Universe website set up by the team from Durham University led by Giles Gasper, Hannah Smithson and  Tom McLeish

I already knew of Grosseteste’s attitude towards natural philosophy and mathematics but didn’t have a suitable quote to hand, so didn’t mention it in my review. Now I do have one. Let us first remind ourselves what Galileo actually said in Il Saggiatore:

Philosophy [i.e. natural philosophy] is written in this grand book — I mean the Universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.

And now what Grosseteste wrote four hundred years earlier in his De lineis, angulis et figuris (On lines, angles and figures) between 1220 and 1235:

“…a consideration of lines, angles and fugures is of the greatest utility because it is impossible to gain a knowledge of natural philosophy without them…for all causes of natural effects must be expressed by means of lines, angles and figures”

Remarkably similar is it not!

 

 

 

 

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

As stated earlier the predominant medieval view of the cosmos was an uneasy bundle of Aristotle’s cosmology, Ptolemaic astronomy, Aristotelian terrestrial mechanics, which was not Aristotle’s but had evolved out of it, and Aristotle’s celestial mechanics, which we will look at in a moment. As also pointed out earlier this was not a static view but one that was constantly being challenged from various other models. In the early seventeenth century the central problem was, having demolished nearly all of Aristotle’s cosmology and shown Ptolemaic astronomy to be defective, without however yet having found a totally convincing successor, to now find replacements for the terrestrial and celestial mechanics. We have looked at the development of the foundations for a new terrestrial mechanics and it is now time to turn to the problem of a new celestial mechanics. The first question we need to answer is what did Aristotle’s celestial mechanics look like and why was it no longer viable?

The homocentric astronomy in which everything in the heavens revolve around a single central point, the earth, in spheres was created by the mathematician and astronomer Eudoxus of Cnidus (c. 390–c. 337 BCE), a contemporary and student of Plato (c. 428/27–348/47 BCE), who assigned a total of twenty-seven spheres to his system. Callippus (c. 370–c. 300 BCE) a student of Eudoxus added another seven spheres. Aristotle (384–322 BCE) took this model and added another twenty-two spheres. Whereas Eudoxus and Callippus both probably viewed this model as a purely mathematical construction to help determine planetary position, Aristotle seems to have viewed it as reality. To explain the movement of the planets Aristotle thought of his system being driven by friction. The outermost sphere, that of the fixed stars drove the outer most sphere of Saturn, which in turn drove the next sphere down in the system and so on all the way down to the Moon. According to Aristotle the outermost sphere was set in motion by the unmoved mover. This last aspect was what most appealed to the churchmen of the medieval universities, who identified the unmoved mover with the Christian God.

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During the Middle Ages an aspect of vitalism was added to this model, with some believing that the planets had souls, which animated them. Another theory claimed that each planet had its own angel, who pushed it round its orbit. Not exactly my idea of heaven, pushing a planet around its orbit for all of eternity. Aristotelian cosmology said that the spheres were real and made of crystal. When, in the sixteenth century astronomers came to accept that comets were supralunar celestial phenomena, and not as Aristotle had thought sublunar meteorological ones, it effectively killed off Aristotle’s crystalline spheres, as a supralunar comet would crash right through them. If fact, the existence or non-existence of the crystalline spheres was a major cosmological debate in the sixteenth century. By the early seventeenth century almost nobody still believed in them.

An alternative theory that had its origins in the Middle Ages but, which was revived in the sixteenth century was that the heavens were fluid and the planets swam through them like a fish or flew threw them like a bird. This theory, of course, has again a strong element of vitalism. However, with the definitive collapse of the crystalline spheres it became quite popular and was subscribed to be some important and influential thinkers at the end of the sixteenth beginning of the seventeenth centuries, for example Roberto Bellarmino (1542–1621) the most important Jesuit theologian, who had lectured on astronomy at the University of Leuven in his younger days.

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Robert Bellarmine artist unknown Source: Wikimedia Commons

It should come as no surprise that the first astronomer to suggest a halfway scientific explanation for the motion of the planets was Johannes Kepler. In fact he devoted quite a lot of space to his theories in his Astronomia nova (1609).

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Astronomia Nova title page Source: Wikimedia Commons

That the periods between the equinoxes and the solstices were of unequal length had been known to astronomers since at least the time of Hipparchus in the second century BCE. This seemed to imply that the speed of either the Sun orbiting the Earth, in a geocentric model, or the Earth orbiting the Sun, in a heliocentric model, varied through out the year. Kepler calculated a table for his elliptical, heliocentric model of the distances of the Sun from the Earth and deduced from this that the Earth moved fastest when it was closest to the Sun and slowest when it was furthest away. From this he deduced or rather speculated that the Sun controlled the motion of the Earth and by analogy of all the planets. The thirty-third chapter of Astronomia nova is headed, The power that moves the planets resides in the body of the sun.

His next question is, of course, what is this power and how does it operate? He found his answer in William Gilbert’s (1544–1603) De Magnete, which had been published in 1600.

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William Gilbert Source: Wikimedia Commons

Kepler speculated that the Sun was in fact a magnet, as Gilbert had demonstrated the Earth to be, and that it rotated on its axis in the same way that Gilbert believed, falsely, that a freely suspended terrella (a globe shaped magnet) did. Gilbert had used this false belief to explain the Earth’s diurnal rotation.

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It should be pointed out that Kepler was hypothesising a diurnal rotation for the Sun in 1609 that is a couple of years before Galileo had demonstrated the Sun’s rotation in his dispute over the nature of sunspots with Christoph Scheiner (c. 1574–1650). He then argues that there is power that goes out from the rotating Sun that drives the planets around there orbits. This power diminishes with its distance from the Sun, which explains why the speed of the planetary orbits also diminishes the further the respective planets are from the Sun. In different sections of the Astronomia nova Kepler argues both for and against this power being magnetic in nature. It should also be noted that although Kepler is moving in the right direction with his convoluted and at times opaque ideas on planetary motion there is still an element of vitalism present in his thoughts.

Kepler conceived the relationship between his planetary motive force and distance as a simple inverse ratio but it inspired the idea of an inverse squared force. The French mathematician and astronomer Ismaël Boulliau (1605–1694) was a convinced Keplerian and played a central roll in spreading Kepler’s ideas throughout Europe.

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Ismaël Boulliau portrait by Pieter van Schuppen Source: Wikimedia Commons

His most important and influential work was his Astronomia philolaica (1645). In this work Boulliau hypothesised by analogy to Kepler’s own law on the propagation of light that if a force existed going out from the Sun driving the planets then it would decrease in inverse squared ratio and not a simple one as hypothesised by Kepler. Interestingly Boulliau himself did not believe that such a motive force for the planet existed.

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Another mathematician and astronomer, who looked for a scientific explanation of planetary motion was the Italian, Giovanni Alfonso Borelli (1608–1697) a student of Benedetto Castelli (1578–1643) and thus a second-generation student of Galileo.

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Giovanni Alfonso Borelli Source: Wikimedia Commons

Borelli developed a force-based theory of planetary motion in his Theoricae Mediceorum Planatarum ex Causius Physicis Deductae (Theory [of the motion] of the Medicean planets [i.e. moons of Jupiter] deduced from physical causes) published in 1666. He hypothesised three forces that acted on a planet. Firstly a natural attraction of the planet towards the sun, secondly a force emanating from the rotating Sun that swept the planet sideway and kept it in its orbit and thirdly the same force emanating from the sun pushed the planet outwards balancing the inwards attraction.

The ideas of both Kepler and Borelli laid the foundations for a celestial mechanics that would eventually in the work of Isaac Newton, who knew of both theories, produced a purely force-based mathematical explanation of planetary motion.

 

 

 

 

 

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

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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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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 Spain in 1551, 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),

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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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