Meeting Index

Eric Voegelin Society Meeting 2009

Western Scientific Pioneers on Eastern Shoulders: From Ibn al-Shatir and al-Khwarizmi to Copernicus and Galileo

Copyright 2003 Peter von Sivers


Overview: Scientific Maturation and Acculturation Processes

          In Greek Antiquity the philosopher Plato sought to apply elements of geometry to the understanding of nature. After him geometry became the first fully developed mathematical science but its utility for analyzing nature was questionable. Plato's student Aristotle took the position that geometry was unfit to serve as a vehicle for the analysis of nature. Scholars, so he argued, should study geometry for its own sake but would be unable to apply it to anything else. In his opinion, scientists cannot undertake a quantitative analysis of nature. They can only give a qualitative description of what they observe.

         [A majority of scholars in Antiquity adopted Aristotle's division between the quantitative- mathematical sciences and the qualitative natural sciences. Only a minority resisted, foremost among them Archimedes and Ptolemy. Archimedes' application of geometry to the study of physics, especially of the behavior of levers, weights, and floating bodies, flatly contradicted Aristotle. Subsequent scholars in Antiquity, however, were too awed by Aristotle and rejected Archimedean physics. The case of Ptolemy was somewhat different. He made a strong effort to harmonize his own geometrical astronomy with descriptive Aristotelian astronomy. As a result, Antiquity ended with a mathematization of the planetary system on Aristotelian terms, while the rest of the natural sciences remained qualitative and descriptive.]

            The Arabs conquered Greek and Indian centers of scientific learning in 635-711. Two centuries later, Muslims had acculturated to the learning of these centers. They embraced both the Aristotelian division between the qualitative and quantitative sciences, as well as Ptolemy's astronomy. Arab and Persian scholars made important contributions to both mathematics and astronomy, improving on Ptolemy's astronomy and pioneering algebra as a new branch of mathematics. Of course, Muslims also added substantially to the qualitative-descriptive sciences of physics, optics, medicine, and alchemy. By contrast, Archimedes' treatise on floating bodies remained untranslated and his non-Aristotelian quantitative physics remained relatively undeveloped in Islamic civilization. Muslims were active in the sciences for about 600 years, especially in mathematics and astronomy, and it was only after c. 1500 that their contributions became more modest.

            Western Christian civilization acculturated, in turn, to the Islamic sciences in the period after 1100 at a time when urbanization and higher institutions of learning were still in their infancy. Although, a century later, scholars had translated most major works from Arabic into Latin, few Europeans possessed the educational preparation to move beyond Aristotle into the fields of mathematics and astronomy. Remarkably, however, Aristotelianism became more central in Western Christian civilization than it ever was in Islamic civilization. Scholastic Aristotelianism or "school knowledge" dominated the university curricula all over Western Europe, culminating in the great commentaries and elaborations of Siger of Brabant and Thomas Aquinas. By 1450, however, many scholars were ready to push beyond Aristotle and delve into the more abstract fields of geometry, algebra, and Ptolemaic astronomy inherited from the Muslims. The discovery of the Greek originals, as well as later Hellenistic and Byzantine manuscripts, inaugurated the Renaissance and provided scholars with the critical wherewithal to probe new scientific frontiers.

            Scholars in Islamic civilization pursued the descriptive and mathematical sciences simultaneously from its beginnings in the 800s. Western Christian civilization, on the other hand, went through two distinctive stages. Scholars began in the 1100 with the qualitative-descriptive sciences and only later, around 1450, turned to the quantitative-mathematical sciences, that is, Ptolemaic astronomy and Archimedean physics. The transition from the first to the second stage was accompanied by considerable intellectual tensions. Many Renaissance scholars were openly hostile to scholastic Aristotelianism. It is in this hostility that we have to look for the origins of the Scientific Revolution of 1500-1700. Ambitious scholars reached a point where they were no longer satisfied with Aristotle=s qualitative observations and descriptions. They wanted to proceed from the description of nature to its mathematization – a critical step in the history of science and technology. By acculturating to the qualitative-descriptive and quantitative-mathematical sciences simultaneously, the Muslims viewed Aristotle far less antagonistically as Renaissance Christians did.

            Two specific, Muslim-pioneered mathematical achievements prepared the ground for Christian Western European scholars to depart from Aristotle and embark on their early modern Scientific Revolution. One achievement was the concept of double circles for expressing planetary motion in astronomy and the other was the generalization of mathematical ratios for both magnitudes and numbers.  Ibn al-Shatir's double circles made astronomy conform more closely to Aristotle than Ptolemy's constructions did. Al-Khwarizmi's integration of geometry and algebra resulted in the foundation of basic mathematics in a much more comprehensive sense than Aristotle had envisaged. Neither scholar, however, felt any urge to overcome Aristotle. For them, Aristotle's overall classification of the sciences as either qualitative-descriptive or quantitative-mathematical continued to be valid.

  Copernicus' Heliocentric Astronomy, 1512

           The Hellenistic astronomer Ptolemy was the first to devise a mathematically rigorous planetary model. He did so, however, within the framework of Aristotle's descriptive geocentric astronomy of the planetary universe. In this universe, the moon, sun, and then-known five planets all revolve at uniform speeds and in perfectly circular spherical routes around the earth. As Ptolemy was aware, careful empirical observation reveals that in this earth-centered or geocentric Aristotelian system some planets do not seem to observe the principles of uniform motion and circularity. To the eye, they approach and move away as well as slow down, even regress, and then speed up again along their paths, thus seemingly denying Aristotle's principles.

            The solution Ptolemy proposed is that of eccentric circular orbits and epicycles. He held that the majority of the planets move in a combination of large and small circles (Fig. 16:1). The large orbit is eccentric, that is, has a center other than the earth, and thereby accounts for the appearance of planets approaching and moving away. The small epicycle, which the planet describes, has its center on the circumference of the orbit. The epicycle explains the appearance of slowing, regressing, and speeding. Thus, in their combined motions, planets move in a complex corkscrew fashion through the sky.

            Of course, in a strict sense an eccentric center located away from the earth violates Aristotle's concept of a single center. Ptolemy ignored this violation since the distance between earth and the eccentric center is relatively small. Observation of planets from their eccentric centers, however, still does not yield completely uniform planetary motion. Ptolemy therefore introduced the "equant center," an additional point at some distance from both the eccentric center and the earth as the only point from which uniform motion can be observed (Fig. 2). With this equant center, however, he had moved dangerously far away from Aristotelianism. If the earth as a center is trigonometrically useless and has to be replaced by two other centers, then Ptolemy's astronomy is not really a mathematical explanation of Aristotelian geocentrism.

            Ptolemy's multiple centers, obviously incompatible with Aristotelian principles, greatly disturbed later Islamic astronomers. They searched for ways to bring Ptolemy closer to Aristotle. In the period of 1200-1400, they developed a number of proposals to bridge the gap. Since it is mathematically possible to transfer the trigonometry of centers to that of epicycles, the Arab astronomer `Ali Ibrahim Ibn al-Shatir (1304-1375) turned the trigonometry of the equant into that of an epicyclet. Accordingly, the planet turns on an epicyclet which circles on an epicycle which, in turn, moves on the eccentric orbit (Fig. 3). With the help of these double epicycles for the majority of the planets and, in addition, a similar circle couplet for the moon (Fig. 4), the Islamic astronomers were able to bring the Ptolemaic system closer to the Aristotelian principles than Ptolemy himself had been able to do. Mathematics could indeed be tweaked to accommodate Aristotle.

            [At present, available records do not indicate that Renaissance scholars knew Ibn al-Shatir's solution for bringing Ptolemy and Aristotle together. The "moon couplet," however, is contained in a Byzantine manuscript which arrived from Constantinople in Rome's Vatican Library sometime in the second half of the 1400s. No reference to this manuscript is found, however, in the known Renaissance texts. No conclusions can be drawn concerning any identifiable readership of the manuscript among Western European astronomers. We know, however, that Copernicus (1473-1543) could read Greek manuscripts and worked in the Vatican Library during his stay in Italy. Remarkably, the Islamic trigonometry of Ibn al-Shatir appears in exactly the same form in the work of Copernicus (Fig. 5), as if there had indeed been a literary transmission. A co-discovery by kindred spirits, of course, is just as plausible and so the question of influences remains for the moment unanswered.]

            Just as his Islamic predecessors, Copernicus used the double planetary epicycles and lunar couplets to explain the seemingly non-uniform movement of the planets since he was a firm believer in Aristotelian uniform motion and circular spheres. Of course, the one, decisive innovation in his astronomical work is the substitution of the sun for the earth at the center of his planetary system. It is here, and here only, that Copernicus shows himself as an anti-Aristotelian. Thus, contrary to widespread belief, Copernicus' revolutionary heliocentrism was not really the long overdue simplification of a hopelessly complex Ptolemaic geocentrism. It was a hybrid construction, with the sophisticated Shatirian double epicycles representing the fully mature Ptolemaic system together with the simpler anti-Aristotelian heliocentric hypothesis. Thus, in the first half of the 1500s, anti-Aristotelianism was not yet a universally accepted intellectual position.

 Galileo's Mathematical Physics, 1604

          Aristotle was a younger contemporary of Eudoxus (c. 408-355). This mathematician was perhaps the most important pioneer of geometry whose formulations entered Euclid's Elements half a century later. It was, in part, because of Eudoxus' rigorous geometrical definitions that Aristotle arrived at his conviction of the unbridgeable gap between qualitative-descriptive physics and quantitative-mathematical geometry. The Elements turned geometry into a self-contained, axiomatic scientific field. It focuses almost entirely on geometrical magnitudes and, with the exception of book VII, has little to say about numbers. For the science of numbers – that is, algebra – to develop, Greeks and Romans had to await the Hellenistic scholar Diophantus (fl. 250 C.E.). The algebra of Diophantus, however, was hampered by the absence of a positional, decimal numerical system. Only once Muslims became familiar with decimal and positional Indian mathematics, in the 800s, did algebra take off.

        The great founding figure of algebra is Muhammad Ibn Musa al-Khwarizmi (c. 780-850), who was a fellow at the caliphally sponsored academy of Baghdad, the House of Wisdom. He is reported to have been the first to write "on the solution of mathematical problems through al-jabr and al-muqabala." The term al-jabr in its literal sense means "reduction" and stands, of course, in its wider sense for algebra, that is, the mathematical field in which powers, roots, fractions, and negative numbers are reduced to positive numbers. The term al-muqabala means "equation" and is as such, of course, the centerpiece of all algebraic calculations. The most important later mathematician was `Umar Khayyam (fl. 1050), who investigated cubic equations and wrote a detailed commentary entitled the "On the Difficulties of Euclid," especially his difficult distinctions between numerical and magnitudinal definitions of proportion in books V and VII. Eventually, an algebra developed which gave numbers an equal status with geometric magnitudes as the basic units mathematicians are dealing with.

            Arab decimal and positional algebra came to Western Europe primarily through the Pisan Leonardo Fibonacci (c. 1170-1240) but his works attracted a large readership only at the end of the 1400s when Diophantus was rediscovered. Euclid's Elements had been translated from both Arabic into Latin in the second half of the1100s. This translation, however, was garbled in the crucial sections on proportions, making it appear as if Euclid had presented only arithmetical definitions in his work on geometry. An accurate translation of Euclid from Greek into Italian – with a correct section on geometrical proportions in book V – eventually appeared only in 1543. Thus, only one generation before Galileo Western Europeans finally had access to the geometry and algebra as the Greeks and Muslims had developed them.

            Like the Muslims before them, sixteenth-century Renaissance mathematicians abandoned the strict Greek and early medieval Christian distinction between discrete arithmetical numbers and continuous geometric magnitudes. The ability to ignore the distinction was the decisive step which allowed for a mathematization of physics. Archimedes, the one Hellenistic scholar who had bucked the trend toward descriptive natural sciences in Antiquity, was of special importance to Renaissance mathematicians interested in physics. His treatise on floating bodies – translated from the Greek – had been available since the late 1200s but attracted the attention of mathematicians only after a new edition came out in 1544. Inspired by this treatise, Italian scholars realized that bodies made of the same material but of different weights fall through a medium (water, air) at the same speed. It was right at this time that they also acquired full access to Greek and Islamic geometry and algebra. The conditions were ripe for questioning the usefulness of an Aristotelian descriptive physics which asserted that bodies of different weights fall at different speeds.

            [The scholar who pioneered the application of mathematics to physics was Galileo (1564-1642). His father was a recognized musical theorist and a musician at the court of the Medici. Galileo himself was an accomplished lutenist who participated in his father's musicological experiments prior to his first appointment as a mathematics professor in Pisa. In one of these experiments, his father used weights to increase and decrease the tension of strings stretched over a sounding box. Galileo discovered that when he made a weight swing back and forth like a pendulum, it behaved like a plucked string.  At the beginning of an experiment, the tone of a string was loud and the pendulum swing was wide. As the experiment continued, the tone became softer but remained the same. Similarly, the pendulum swing became shorter but the time required for each was unchanged. Aristotle, so Galileo concluded, was incorrect when he declared upward and downward motions to be the only natural movements of bodies on earth. There was a third, pendulous movement in nature.]

            Around 1604, at the beginning of his university appointment at Padua, Galileo developed an interest in another form of bodily movement, that of the accelerated fall. Precise experiments with falling bodies were difficult to conceive. Dropping weights from the tower of Pisa, contrary to widespread belief, taught nothing since the observers could not be in two places at once. So Galileo used inclined beams as approximations. A craftsman built him a c. 6½-foot long grooved beam, propped up at a slight angle, on which brass balls could be rolled down. As timing device, Galileo used a water container with a drip hole and a scale for weighing the drops.

            Using his observation of equal times but varying swing widths in pendulums as a hunch, Galileo decided to divide the total time of the ball's accelerating descent down the beam into eight equal time intervals. He placed a marker at each point where the ball completed one of the eight time intervals. As he measured the distances between the markers he discovered that their increasing lengths roughly followed the progression of the odd integers 1, 3, 5, 7, 9, 11, 13, and 15. Ignoring the apparent incompatibility between magnitudes and numbers, Galileo realized that the numbers of these distances, if added consecutively, formed squares which were actually the squares of the cumulative time intervals. For example, 1+3=4=2x2, 1+3+5=9=3x3, 1+3+5+7=16=4x4, etc. Generalized, s=txt, where s stands for distance and t for time. Subsequently, Galileo discovered that the same ratios applied inversely to the lengths of pendulums and the decreasing width of their swings. Galileo concluded that, contrary to Aristotle, earthly motion – such as the accelerating movement of bodies falling to the ground – could indeed be mathematized.

            [As Galileo was later to realize, he had forerunners. In the fourteenth century, professors at the Universities of Oxford and Paris had reflected on Aristotle's discussion on "the intension and remission of forms and qualities," such as changing colors, temperatures, or speeds of bodies. Even given the limited mathematics of the period, these professors proposed geometrical ratios between distances and squares of time for bodies in accelerating motion as Galileo did. They remained, however, firmly within the Aristotelian framework of qualities. Even though they came de facto upon the law of falling bodies, by viewing it in analogy to changing colors and temperatures they remained firmly rooted in Aristotelian descriptive-qualitative science. Furthermore, they did not carry out experiments or specified how they imagined uniform acceleration would take place. Thus, their proposal an remained abstract, hypothetical thought-play. Only Galileo can be called the anti-Aristotelian scientific revolutionary who laid the foundation for the mathematization of physics.]

  General Conclusions

          (1) Muslims acculturated to the inherited parallelism between qualitative-descriptive natural and quantitative-analytical mathematical sciences from the start. The Islamic empire was the heir of a highly sophisticated educational infrastructure, extending from former Byzantine and Sasanid Persian to northern Indian cities. After a 200 year learning period, the Arabs were at home in the natural and mathematical sciences just as much as their Greeks, Persian, and Indians predecessors.

            (2) Medieval Christians inherited a delapidated Western Roman empire with a minimal educational infrastructure. Not surprisingly, when they began to rebuild this infrastructure in the 1100s – mostly thanks to the Islamic heritage in Spain and Sicily – they began with the basics, that is, the qualitative-descriptive sciences. Only once they had mastered these sciences did they turn in c. 1450 to the quantitative-mathematical sciences. Since Aristotle had drawn a dogmatic line between the two sets of science, Renaissance Christians challenged Aristotelian dogma and redrew the line by mathematizing astronomy and physics.

            (3) The parallelism between qualitative-descriptive and quantitative-mathematical sciences characterizes all civilizations. In a more general sense, of course, this parallelism extends to the humanities versus the sciences. The line between the two is fuzzy and hence subject to debate. What was Aristotelian dogmatism in the Middle Ages is today reductionism or scientism as the frame within which the debate takes place.

            (4) There have been three major intellectual jolts in world history which have shaken scholars into reexamining the dividing line between the humanities and sciences. One occurred in Greece when Plato declared geometry the science without which no philosopher should enter the Academy. A second one occurred in Renaissance Poland and Italy when Copernicus and Galileo mathematized astronomy and physics, as discussed in this paper. A third one occurred in the 1920s when Niels Bohr and Werner Heisenberg, with quantum mechanics, reintroduced subjectivity into the sciences and thereby rehabilitated the human sciences with their inalienable qualitative-descriptive orientation.

            (5) Contrary to widespread belief, these jolts were not occurrences in Western civilization alone. As should be uncontested by now, Plato is understandable only within a larger Perso-Greek space which includes sixth-century B.C.E. Zoroastrianism and Judaism. Al-Shatir, al-Khwarizmi, Copernicus, and Galileo belong to a single geographical-intellectual milieu, as I hope I have demonstrated in this paper. Quantum mechanics represents the level of mature quantitative-mathematical sciences in the general post-World War I crisis when the limits of the ideology of progress and the Western hegemony over a colonial world had become visible.

  Introductory Bibliography

          Islamic Science. The study of the transmission of mathematics and Aristotelian sciences to Europe is still largely a field for specialists publishing their research in article form. For astronomy, a layperson will find the short essay of George Saliba, Rethinking the Roots of Modern Science: Arabic Manuscripts in European Libraries (Washington, D.C.: Center for Contemporary Arab Studies at Georgetown University, 1999), most accessible. For mathematics, a number of entries in Roshdi Rashed, Encyclopedia of the History of Arabic Science (London: Routledge, 1996), are recommended, especially those on pp. 376-417.

            Astronomy. Here, the historical literature is immense. The following are or contain accessible, richly illustrated introductory surveys: James McClellan III and Harold Dorn, Science and Technology in World History: An Introduction (Baltimore, Md.: Johns Hopkins University Press, 1999); Michael Hoskin, ed., The Cambridge History of Astronomy (Cambridge: Cambridge University Press, 1999); Ronald Brashear and Daniel Lewis, Star Struck: One Thousand Years of the Art and Science of Astronomy (San Marino, Calif.: The Huntington Library, 2001).

            Physics. The literature is similarly immense. McClellan and Dorn, cited above, present a good summary. The recognized biographer of Galileo is Stillman Drake, Galileo: Pioneer Scientist (Toronto: University of Toronto Press, 1990). See also his Essays on Galileo and the History and Philosophy of Science, selected and introduced by N.M. Swerdlow and T.H. Levere, (Toronto: University of Toronto Press, 1999), 3 vols.

          The brackets in the text ("["and "]") indicate passages possibly to be left out during presentation in Philadelphia. This paper is excerpted and adapted for the Philadelphia 2003 meeting from a chapter in Peter von Sivers, Charles A. Desnoyers, and George B. Stow, Patterns of World History: Origins, Adaptations, Interactions (New York: Longman, planned publication date 2005).