Category Archives: History of Islamic Science

Renaissance Science – VIII

In the last two episodes we have looked at developments in printing and art that, as we will see later played an important role in the evolving Renaissance sciences. Today, we begin to look at another set of developments that were also important to various areas of the newly emerging practical sciences, those in mathematics. It is a standard cliché that mathematisation played a central roll in the scientific revolution but contrary to popular opinion the massive increase in the use of mathematics in the sciences didn’t begin in the seventeenth century and certainly not as the myth has it, with Galileo.

Medieval science was by no means completely devoid of mathematics despite the fact that it was predominantly Aristotelian, and Aristotle thought that mathematics was not scientia, that is, it did not deliver knowledge of the natural world. However, the mathematical sciences, most prominent astronomy and optics, had a fairly low status within medieval university culture.

One mathematical discipline that only really became re-established in Europe during the Renaissance was trigonometry. This might at first seem strange, as trigonometry had its birth in Greek spherical astronomy, a subject that was taught in the medieval university from the beginning as part of the quadrivium. However, the astronomy taught at the university was purely descriptive if not in fact even prescriptive. It consisted of very low-level descriptions of the geocentric cosmos based largely on 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. Mathematical astronomy requiring trigonometry was not hardy taught and rarely practiced.

Both within and outside of the universities practical astronomy and astrology was largely conducted with the astrolabe, which is itself an analogue computing device and require no knowledge of trigonometry to operate.

Before we turn to the re-emergence of trigonometry in the medieval period and its re-establishment during the Renaissance, it pays to briefly retrace its path from its origins in ancient Greek astronomy to medieval Europe.

The earliest known use of trigonometry was in the astronomical work of Hipparchus, who reputedly had a table of chords in his astronomical work. This was spherical trigonometry, which uses the chords defining the arcs of circles to measure angles. Hipparchus’ work was lost and the earliest actual table of trigonometrical chords that we know of is in Ptolemaeus’ Mathēmatikē Syntaxis or Almagest, as it is usually called today.

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

When Greek astronomy was appropriated in India, the Indian astronomers replaced Ptolemaeus’ chords with half chords thus creating the trigonometrical ratios now known to us, as the sine and the cosine.

It should be noted that the tangent and cotangent were also known in various ancient cultures. Because they were most often associated with the shadow cast by a gnomon (an upright pole or post used to track the course of the sun) they were most often known as the shadow functions but were not considered part of trigonometry, an astronomical discipline. So-called shadow boxes consisting of the tangent and cotangent used for determine heights and depths are often found on the backs of 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

  Islamic astronomers inherited their astronomy from both ancient Greece and India and chose to use the Indian trigonometrical half chord ratios rather than the Ptolemaic full cords. Various mathematicians and astronomers made improvements in the discipline both in better ways of calculating trigonometrical tables and producing new trigonometrical theorems. An important development was the integration of the tangent, cotangent, secant and cosecant into a unified trigonometry. This was first achieved by al-Battãnī (c.858–929) in his Exhaustive Treatise on Shadows, which as its title implies was a book on gnonomics (sundials) and not astronomy. The first to do so for astronomy was Abū al-Wafā (940–998) in his Almagest.

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Image of Abū al-Wafā Source: Wikimedia Commons

It was this improved, advanced Arabic trigonometry that began to seep slowly into medieval Europe in the twelfth century during the translation movement, mostly through Spain. It’s reception in Europe was very slow.

The first medieval astronomers to seriously tackle trigonometry were the French Jewish astronomer, Levi ben Gershon (1288–1344), the English Abbot of St Albans, Richard of Wallingford (1292–1336) and the French monk, John of Murs (c. 1290–c. 1355) and a few others.

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Richard of Wallingford Source: Wikimedia Commons

However, although these works had some impact it was not particularly widespread or deep and it would have to wait for the Renaissance and the first Viennese School of mathematics, Johannes von Gmunden (c. 1380­–1442), Georg von Peuerbach (1423–1461) and, all of whom were Renaissance humanist scholars, for trigonometry to truly establish itself in medieval Europe and even then, with some delay.

Johannes von Gmunden was instrumental in establishing the study of mathematics and astronomy at the University of Vienna, including trigonometry. His work in trigonometry was not especially original but displayed a working knowledge of the work of Levi ben Gershon, Richard of Wallingford, John of Murs as well as John of Lignères (died c. 1350) and Campanus of Novara (c. 200–1296). His Tractatus de sinibus, chordis et arcubus is most important for its probable influence on his successor Georg von Peuerbach.

Peuerbach produced an abridgement of Gmunden’s Tractatus and he also calculated a new sine table. This was not yet comparable with the sine table produced by Ulugh Beg (1394–1449) in Samarkand around the same time but set new standards for Europe at the time. It was Peuerbach’s student Johannes Regiomontanus, who made the biggest breakthrough in trigonometry in Europe with his De triangulis omnimodis (On triangles of every kind) in 1464. However, both Peuerbach’s sine table and Regiomontanus’ De triangulis omnimodis would have to wait until the next century before they were published. Regiomontanus’ On triangles did not include tangents, but he rectified this omission in his Tabulae Directionum. This is a guide to calculating Directions, a form of astrological prediction, which he wrote at the request for his patron, Archbishop Vitéz. This still exist in many manuscript copies, indicating its popularity. It was published posthumously in 1490 by Erhard Ratdolt and went through numerous editions, the last of which appeared in the early seventeenth century.

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A 1584 edition of Regiomontanus’Tabulae Directionum Source

Peuerbach and Regiomontanus also produced their abridgement of Ptolemaeus’ Almagest, the Epitoma in Almagestum Ptolemae, published in 1496 in Venice by Johannes Hamman. This was an updated, modernised version of Ptolemaeus’ magnum opus and they also replaced his chord tables with modern sine tables. A typical Renaissance humanist project, initialled by Cardinal Basilios Bessarion (1403–1472), who was a major driving force in the Humanist Renaissance, who we will meet again later. The Epitoma became a standard astronomy textbook for the next century and was used extensively by Copernicus amongst others.

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Title page Epitoma in Almagestum Ptolemae Source: Wikimedia Commons

Regiomontanus’ De triangulis omnimodis was edited by Johannes Schöner and finally published in Nürnberg in 1533 by Johannes Petreius, together with Peuerbach’s sine table, becoming a standard reference work for much of the next century. This was the first work published, in the European context, that treated trigonometry as an independent mathematical discipline and not just an aide to astronomy.

Copernicus (1473–1543,) naturally included modern trigonometrical tables in his De revolutionibus. When Georg Joachim Rheticus (1514–1574) travelled to Frombork in 1539 to visit Copernicus, one of the books he took with him as a present for Copernicus was Petreius’ edition of De triangulis omnimodis. Together they used the Regiomontanus text to improve the tables in De revolutionibus. When Rheticus took Copernicus’ manuscript to Nürnberg to be published, he took the trigonometrical section to Wittenberg and published it separately as De lateribus et angulis triangulorum (On the Sides and Angles of Triangles) in 1542, a year before De revolutionibus was published.

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Rheticus’ action was the start of a career in trigonometry. Nine years later he published his Canon doctrinae triangvlorvmin in Leipzig. This was the first European publication to include all of the six standard trigonometrical ratios six hundred years after Islamic mathematics reached the same stage of development. Rheticus now dedicated his life to producing what would become the definitive work on trigonometrical tables his Opus palatinum de triangulis, however he died before he could complete and publish this work. It was finally completed by his student Valentin Otto (c. 1548–1603) and published in Neustadt and der Haardt in 1596.

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Source: Wikimedia Commons

In the meantime, Bartholomäus Piticus (1561–1613) had published his own extensive work on both spherical and plane trigonometry, which coined the term trigonometry, Trigonometria: sive de solution triangulorum tractatus brevis et perspicuous, one year earlier, 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, whereas those of Rheticus were calculated up to more than twenty places for large angles and fifteenth for small ones. In comparison Peuerbach’s sine tables from the middle of the fifteenth century were only accurate to three places of decimals. 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.

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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.

In the seventeenth century a major change in trigonometry took place. Whereas throughout the Renaissance it had been handled as a branch of practical mathematics, used to solve spherical and plane triangles in astronomy, cartography, surveying and navigation, the various trigonometrical ratios now became mathematical functions in their own right, a branch of purely theoretical mathematics. This transition mirroring the general development in the sciences that occurred between the Renaissance and the scientific revolution, from practical to theoretical science.

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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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The House of Wisdom is a Myth

When I first got really interested in the history of science, the history of science of the Islamic empires was not something dealt with in any detail in general works on the topic. If you wanted to get to know anything much about what happened in the various areas of the world dominated by Islamic culture in the period between the seventh and sixteenth centuries then you had to find and read specialist literature produced by experts such as Edward Kennedy. Although our knowledge of that history still needs to be improved, the basic history has now reached the popular market and people can inform themselves about major figures writing in Arabic on various areas of science between the demise of classical antiquity and the European Renaissance such as the mathematician Muḥammad ibn Mūsā al-Khwārizmī, the alchemist Abū Mūsā Jābir ibn Hayyān, the optician, Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham or the physician Abū Bakr Muhammad ibn Zakariyyā al-Rāzī. These and a handful of other ‘greats’ are not as well known as their later European counterparts but knowledge of them, usually under their popular names, so al-Khwarizmi, Jabir, al-Haytham and al-Razi, is these days quite widespread amongst well educated and well read people. There is even a flourishing popular book market for titles about Islamic science.

Amongst those non-professionals, who interest themselves for the topic, particularly well known is the so-called House of Wisdom, a reputed major centre for scientific translation and research in Baghdad under the Abbasid Caliphs. This reputed academic institution even provided the title for two of the biggest selling popular books on Islamic science Jim al-Khalili’s The House of Wisdom: How Arabic Science Saved Ancient Knowledge and Gave Us the Renaissance and Jonathan Lyons’ The House of Wisdom: How the Arabs Transformed Western Civilisation. Neither Jim al-Khalili nor Jonathan Lyons is a historian of science, let alone Islamic science; al-Khalili is a physicist and broadcaster and Lyons is a journalist and herein lies the rub. Real historians of Islamic science say that the House of Wisdom never existed, at least not in any form remotely resembling the institution presented by al-Khalili, Lyons and other popular sources including, unfortunately Wikipedia, where the article is largely based on Lyons’ pop book.

The picture painted by al-Khalili and Lyons, and to be fair they didn’t create it but copied it from other fantasts, is of a special academic research institution set up by the early Abbasid Caliphs, staffed with leading scientific scholars, who carried out a sponsored programme of translating Greek scientific texts, which they them analysed, commented and developed further. Here academic exchanges, discussions, conferences took place amongst the leading scientific scholars in the Abbasid Empire.

The reality looks very different.[1]To quote Gutas (page 54):

It is in this light that the very scanty reliable reports about the bayt al-hikmashould be evaluated. Much ink has been used unnecessarily on description of the bayt al-hikma, mostly in fanciful and sometimes wishful projections of modern institutions and research projects back into the eighth century. The fact is that we have exceedingly little historical [emphasis in original] information about the bayt al-hikma. This in tself would indicate that it was not something grandiose or significant, and hence a minimalist interpretation would fit the historical record better.

The bayt al-hikma, to give it its correct name, which doesn’t really translate as house of wisdom, was the palace archive and library or repository, a practice taken over by the Abbasid Caliphs from the earlier Sassanian rulers along with much other royal court procedure to make their reign more acceptable to their Persian subjects. The wisdom referred to in the translation refers to poetic accounts of Iranian history, warfare, and romance. The Abbasid Caliphs appear to have maintained this practice now translating Persian historical texts from Persian into Arabic. There is absolutely no evidence of Greek texts, scientific or otherwise, being translated in the bayt al-hikma.

Much is made of supposed leading Islamic scientific scholars working in the bayt al-hikmaby the al-Khalili’s, Lyons et al. In fact the first librarian under the Abbasids was a well-known Persian astrologer, again a Sassanian practice taken over by the Abbasids. Later al-Khwarizmi and Yahya ibn Abi Mansur both noted astronomers but equally noted astrologers served in the bayt al-hikmaunder the Abbasid Caliph al-Ma’mun.

We will give Gutas the final word on the subject (page 59):

The bayt al-hikmawas certainly also not an “academy” for teaching the “ancient” sciences as they were being translated; such a preposterous idea did not even occur to the authors of the spurious reports about the transmission of the teaching of these sciences that we do have. Finally it is not a “conference centre for the meeting of scholars even under al-Ma’mun’s sponsorship. Al-Ma’mun, of course (and all the early Abbasid caliphs), did host scholarly conferences or rather gatherings, but not in the library; such gauche social behaviour on the part of the caliph would have been inconceivable. Sessions (magalis) were held in the residences of the caliphs, when the caliphs were present, or in private residences otherwise, as the numerous descriptions of them that we have indicate.

As a final comment we have the quite extraordinary statement made by Jim al-Khalili on the BBC Radio 4 In Our Time discussion on Maths in the Early Islamic World:

In answer to Melvyn Braggs question, “What did they mean by the House of Wisdom and what sort of house was it? It is supposed to have lasted for 400 years, it is contested”

Jim al-Khalili: “It is contested and I’ll probably get into hot water with historians but let’s say what I think of it. There was certainly potentially something called the house of wisdom a bit like the Library of Alexandria many centuries earlier, which was a place where books were stored it may have also been a translation house. It was in Baghdad this was in the time of al-Ma’mun, it may have existed in some form or other in his father’s palace…”

Bragg: “Was it a research centre, was it a place where people went to be paid by the caliphs to get on with the work that you do in mathematics?”

Al-Khalili: “I believe it very well could have been…” He goes on spinning a fable, drawing parallels with the Library of Alexandria

History is not about what you choose to believe but is a fact-based discipline. Immediately after al-Khalili’s fairy story Peter Pormann, Professor of Classics & Graeco-Arabic Studies at the University of Manchester chimes in and pricks the bubble.

Pormann: “There’s the myth of the House of Wisdom as this research school, academy and so on and so forth, basically there is very little evidence…”

Listen for yourselves!

I find Bragg’s choice of words, repeated by al-Khalili, “it is contested” highly provocative and extremely contentious. It is not contested; there is absolutely no evidence to support the House of Wisdom myth as presented by Lyons, al-Khalili et al. What we have here is another glaring example of unqualified pop historians propagating a myth and blatantly ignoring the historical facts, which they find boring.

[1]The facts in the following are taken from Dimitri Gutas, Greek Thought, Arabic Culture: The Graeco-Arabic Translation Movement in Baghdad and Early Abbasid Society (2nd–4th/8th–10th centuries), Routledge, Oxford, ppb. 1998 pp. 53-60 and Lutz Richter-Bernburg, Potemkin in Baghdad: The Abbasid “House of Wisdom” as Constructed by 1001 inventions In Sonja Brentjes–Taner Edis­–Lutz Richter-Bernburg eds., 1001 Distortions: How (Not) to Narrate History of Science, Medicine, and Technology in Non-Western Science, Biblioteca Academica Orientalistik, Band 25, Ergon Verlag, Würzburg, 2016 pp. 121-129

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