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## 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

[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.

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

*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

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.]

(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.

*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.

*Patterns of World History: Origins, Adaptations, Interactions* (New
York: Longman, planned publication date 2005).