More importantly: I wrote this article before I had read and
digested
the work of Jens Høyrup. Høyrup makes the valid
distinction
in pre-modern times between "sub-scientific" and "scientific"
mathematics.
The former is practitioners' mathematics, developed independent of its
"scientific" counterpart. This differs fundamentally from our
modern
distinction between "applied" and "pure" mathematics. See in
particular
his "Sub-Scientific Mathematics. Observations on a Pre-Modern
Phenomenon"
History of Science 28 (1990), pp. 63-86 (Reprinted in his book __In
Measure, Number, and Weight:__

__Studies in Mathematics and Culture__. Albany: State
University of New York Press, 1994.)

Also, I was still influenced by Roshdi Rashed when I wrote
this. It turns out that al-Khwarizmi did NOT invent
algebra. It was already practiced by trade groups and was
transmitted orally before al-Khwarizmi's time.

**Introduction**

In the next twenty minutes I will attempt to put into political and
social perspective the mathematics done in the ninth (Christian)
century
in the Islamic world. I have chosen to concentrate on the figure
of al-Khwarizmi, mainly because he is fairly well-known, and because
his
*Algebra*
exhibits a particular anomaly to the modern reader. To rectify
common
misconceptions about Islamic science, and about the famous House of
Wisdom
in particular, I find it necessary to spend some time discussing
astrology, whose undercurrents were present in every area of foreign
intellectual
enquiry of the time.

**The apparent anomaly in the Algebra of al-Khwarizmi**

The *Book of al-jabr (restoring) and al-muqabala (balancing)*
(henceforth
*Algebra*)
was written by al-Khwarizmi in Baghdad sometime during the reign of the
‘Abbasid caliph al-Ma’mun (reigned 813-833). Reading the first
half
of the book one delves into the topic of algebra as it existed on its
inception.
Rules are presented which instruct the reader on the solution to any
algebraic
problem in one variable which is reducible to a quadratic equation.

Then come the applications, which occupy nearly half the book.
There are problems on mercantile transactions, geometry, and
testimonies.
It is the examples involving the distribution of estates which take up
the bulk of the applications. Islamic law, as recorded in the
Qu’ran,
offers very complicated rules for the distribution of estates.
This
accounts for the weight given to such problems in the *Algebra*.

Were this book a work of pure mathematics, we might expect a few
pseudo-applied
problems, like the ones encountered in the related rates section of a
calculus
book. But there are so many different problems worked out on
inheritances
alone that we might declare the work to be a handbook for judges who
are
involved with such cases. So tentatively we might say that
al-Khwarizmi’s
*Algebra
*is
just what the author says it is in his introduction, a “work on
algebra,
confining it to the fine and important parts of its calculations, such
as people constantly require in cases of inheritance, legacies,
partition,
law-suits, and trade, and in all their dealings with one another, or
where
surveying, the digging of canals, geometrical computation, and other
objects
of various sorts and kinds are concerned.” [1]
That sure sounds like the description of a practical handbook.

But without exception, every applied problem in the book is reducible to a linear equation. The theory of quadratic polynomials is never used, despite the fact that al-Khwarizmi spends quite some time explaining and proving quadratic methods. So this is the dilemma: why is it that a single book serves both as a treatise on a new, theoretical subject and as a practical book for non-mathematicians?

**The Greeks**

This applied nature of the *Algebra* certainly makes it unlike
anything produced in classical Greece. You won’t find a word in
Euclid’s
*Elements*
about digging canals or surveying, for example. Greek mathematics
up to the time of Euclid was aristocratic, in the sense that it did not
soil itself with everyday concerns. If math were inspired by
quotidian
activities, this inspiration was not expressed in the texts.
Geometry,
Plato wrote, “is for the sake of knowing what is always, and not at all
for what is at any time coming into being and passing away.” [2],
and applications to geometry are mere “by-products” [3].

For theoretical, proof-based mathematics to develop as it did, it needed to incubate in the immaterial realm of ideal forms or ideas, away from the imprecise and changing real world. Though the connection between mathematics and philosophy in Greece is quite complex [4], it still remains true that lines, points, and geometrical objects in general have no existence in the mundane world for the Greeks.

With the gradual orientalizing of Greek culture following the
conquests
of Alexander, this aversion for applications dissipated somewhat.
Archimedes (d. 212 BC), the great geometer, who was as concerned with
rigorous
proof as any of his predecessors, also built war machines. The
first
sentence in Book VIII of Pappus’ *Collection* (ca. AD 320) reads
like
this: “The science of mechanics, my dear Hermodorus, has many
important
uses in practical life, and is held by philosophers to be worthy of the
highest esteem, and is zealously studied by mathematicians, because it
takes almost first place in dealing with the nature of the material
elements
of the universe.” [5]

But it is not the mathematics of late antiquity which reached
Baghdad
in the eighth century, but mainly the mathematics of Euclid, the
rigorous
geometry of Archimedes, and the distilled ideas of Aristotle and
Plato.
This dichotomy between the squalid, changing earthly realm of real
existence
and the static realm of ideal forms was an idea foreign to Islamic
scientists. [6]
For the Muslims, in the words of Jens Hoyrup, “no practice is too mean
to serve as a startingpoint for theoretical reflection, and no theory
too
lofty to be applied in practice.” [7]
Euclid’s *Elements*, for example, was translated into Arabic in
the
eighth century to serve the practical needs of surveyors and
engineers. [8]

In Greece the mathematical sciences were divided into four
fields:
arithmetic, geometry, astromony, and music. This is the classical *quadrivium*,
which was adopted by Islamic scholars as well, though sometimes with
modifications. [9]
Note in particular that astronomy is a subfield of mathematics.

**Let**’**s examine the **‘**Abbasid intellectual and
political
climate**

The Ummayad dynasty, which ruled from 661 to 750 CE, continued the conquest of the world which was begun by Mohammed and the “Rightly Guided Caliphs”, eventually engaging in war such distant foes as the French and the Chinese. By the end of the seventh century the Sasanian Empire of Persia had been thoroughly defeated. Arabs had no experience with the day-to-day management of a large empire, so the administrative needs of the Islamic state were normally met by leaving intact the local bureaucratic structure which they found in each country. Greek, for instance, remained the language of the administrators of Syria for over a half century after the Arab conquest. [10] In Iran, local Zoroastrian landholders collected taxes for the conquerors now instead of the King. [11] Being the ones in power, Arabs of course had many advantages over subjugated peoples. It was especially the now subaltern Persians, feeling culturally superior to their nomadic Bedouin overlords, who were itching for things to change.

In a well-planned and broadly supported movement, the Ummayads were overthrown in 750 by a new dynasty, the ‘Abbasids. A key source of ‘Abbasid support lay in Khurasan, a region in northeastern Persia. Broad support existed in other regions as well, but this revolt can be seen largely as a re-assertion of Persian national identity, against the pro-Arab Ummayads. In fact, it can be said that Persian culture reemerged in full force, in Islamic garb, under the ‘Abbasids. Al-Khwarizmi himself was of Persian stock, his ancestors coming from Khwarezm, in distant Transoxania. The Banu Musa, al-Mahani, and a host of others in the intellectual circle of ninth century Baghdad, were also Persians.

Iran was the heartland of the toppled Sasanian monarchy, and it was Sasanian imperial ideology, based on Zoroastrian myth, which appealed to the second ‘Abbasid caliph al-Mansur (reigned 754-775). The caliph needed support from the Iranian nobility, mostly still Zoroastrian, so his appropriation of this Persian system was certainly a political move. The important aspect of it which interests us here is the tradition of translation.

The myth goes like this: In the 4th century BC Alexander the Great conquered the Persian empire and deposed its last king, Darius III. Alexander marched on the Persian capital at Persepolis, where he found writings on the various sciences and branches of knowledge—knowledge which was divine in origin, received from Ohrmazd by the prophet Zoroaster himself. Alexander had all these books translated from Persian into Greek and Coptic (Egyptian), after which he burned the originals. Centuries later, in Sasanian times, the kings issued an order for the collection of foreign works, which were to be retranslated back into Persian. In this way the Persian kings justified the adoption of foreign learning, since the books were not really foreign at all, just stolen from Persians, mainly by the Greeks. [12]

This story, first propagated by Sasanid astral historians, was
repeated,
in slightly different versions, by ‘Abbasid astrologers. The
divine
origin of the original Avestan works was later ascribed to other
figures,
most notably to the Egyptian Hermes, legendary founder of the
Greek
occult sciences. [13] Abu
Sahl
al-Fadl Ibn Nawbakht, a Persian astronomer and astrologer under the
caliph
Harun al-Rashid (reigned 786-809), is the author of one of these
histories.
He writes of the ancient libraries as twelve in number, one for each
sign
of the zodiac. David Pingree notes “The zodiacal-planetary
libraries
of science themselves seem to be but imitations of Harun al-Rashid’s
Treasure-house
of Wisdom…” [14] In
fact,
ibn Nawbakht worked at the House of Wisdom, and his *History
*is characteristic
of the activities of that library, as we shall soon see.

Al-Mansur adopted this Persian tradition of translation, though now the works were to be rendered into Arabic, the language on the ascendant. Full caliphal support was offered to the movement. At first treatises in Pahlavi (Middle Persian) were translated into Arabic, and soon after Syriac versions of Greek works, then the Greek originals themselves were sought out. The subjects chosen were practical ones: astrology, the mathematical sciences (geometry, arithmetic, and astronomy), and also agriculture. [15] To this list must be added medical works, translated into Arabic by the doctors of Jundishapur, south of Baghdad. The search for Greek philosophical works began early, also, to aid in the art of disputation. This is because Muslims found themselves unable to defend their faith against the more sophisticated Christians and Jews, who were steeped in the logical works of Aristotle and other Greek philosophers.

**The House of Wisdom**

Many of you have read something about al-Ma’mun’s House of Wisdom,
usually
called in Arabic the *bait al-hikma*, and of al-Khwarizmi’s
association
with it. Most of what has been written about the institution has
been either exaggerated or is just plain wrong. George Makdisi
and
Dimitri Gutas have each helped clear up a lot of the confusion which
surrounds
the nature of this palace library, and I will review here what is
known.

Makdisi, in his seminal book *The Rise of Colleges*, explains
that
the *bait al-hikma* was a library, and that there were several
such
libraries throughout the Islamic world. He writes: “The
activities
that took place in libraries were those involved with books, such as
reading
and copying.” [16]
“These
institutions were libraries essentially; nothing in the sources allows
us to assimilate them as colleges.” [17]

In his book *Greek Thought, Arabic Culture* Gutas devotes nine
pages to shooting down myths surrounding the famous Baghdad library,
and
on proposing a more sound theory of its origins and purpose. The
paucity of information, he writes, “in itself would indicate that
it was not something grandiose or significant…” [18]
In tune with his emphasis on the Persian origins of ‘Abbasid political
ideology, he notes that the Arabic term *bayt al-hikma* is a
translation
of the Sasanian word for “library”, meaning a palace library which
“functioned
as an idealized national archive: as the place where poetic
accounts
of Iranian history, warfare, and romance were transcribed and
preserved…” [19]
He continues: “We have no reason to doubt that in the early ‘Abbasid
administration
it retained this function since its adoption was effected by
individuals
who were carriers of Sasanian culture and under the mandates of a
caliphal
policy to project Sasanian imperial ideology. Its function, in
other
words, was to transcribe and preserve books on *Iranian* national
history, warfare, and romance.” [20]

The library was not the locus of translation activities. The
only
two known references to translation in the institution show that they
were
made from Persian, not Greek, into Arabic. And of the many
references
to translations from the Greek, not one names the *bait al-hikma*.
In addition, Gutas writes: “The *bayt al-hikma* was
certainly
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 was not a ‘conference’
center
for the meetings of scholars even under al-Ma’mun’s sponsorship.” [21]
So it was a library and nothing more. After all, that is what “House of
Wisdom” should mean anyway. I even have an example of an
independent,
modern version of this name: in large brass letters on the wall
of
the library here at the University of Indianapolis is written “Treasury
of Wisdom”. And whoever put that up was not thinking of the
Baghdad
library.

**Astrology, astronomy and mathematics**

In Sasanian Persia, astrology played a fundamental role in the lives of the ruling class. This is in contrast to the Arab Muslims, who were warned by the Prophet himself to avoid study the subject, for it leads to divination. [22] Thus astrology played only a minor role under the Ummayads. [23] It is with the Persian revival under the ‘Abbasids that astrology became an integral part of Islamic court life. Imitating the Sasanian kings, the caliphs from al-Mansur on hired astrologers to cast horoscopes, foretell events, and to glorify their reigns by writing astrological histories.

Astrology experienced its Islamic golden age in the ninth
century.
The caliph and his powerful advisers employed many astrologers, which
resulted
in a demand for astronomy and mathematics. While in Roman
Imperial
times the words for mathematician and astrologer (*mathematicus(-os)*)
were the same, the Arabic word *munajjim *could mean astrologer
or
astronomer. [24]
There
was a large overlap between the three fields. In reviewing the
scholars
listed in GAS volumes V, VI and VII (mathematics, astronomy, and
astrology
respectively), I find that of the roughly 79 people who worked in at
least
one of the three fields in Baghdad in the ninth century, twenty precent
worked in all three. Because we lack information on many of these
men, the actual percentage was probably much higher. Half the
astrologers
also are known to have worked in astronomy, and over three quarters of
the mathematicians were also astronomers. No one is listed as
having
been a mathematician and astrologer, but not an astronomer. Of
course
there are some serious problems with the data, but my point that the
relationship
between mathematics, astronomy and astrology was a strong one is
indisputable. [25]

It was not until the tenth century that serious attacks on astrology appear, both by philosophers and theologians. From this time on most mathematicians and astronomers tended to dissociate themselves from astrology, though they were often pulled back to it by the demands of the court. Al-Khayyam was hired as an astrologer even though he did not believe in astrology, for example. [26]

**More on the House of Wisdom**

About the House of Wisdom Gutas writes “Under al-Ma’mun it appears
to
have gained an additional function related to astronomical and
mathematical
activities; at least this is what the names associated with the *bayt
al-hikma* during that period would imply.” [27]
It is true that five of the six people known to have been associated
with
the library in this time were astronomers or mathematicians, but the
astrological
element is also present. There is no reason to suppose, for
example,
that al-Khwarizmi wrote his *Algebra* as part of his duties at
the
library. In this section I will consider the connection of each
‘librarian’
to the science of astrology.

Under al-Ma’mun the director of the House of Wisdom was Sahl ibn-Harun (d. 830), the great poet, astrologer and translator of Pahlavi literature—not a scientist by any means. [28] The only other people known to have been associated with the library at this time were Yahya ibn-Abi-Mansur, the Banu Musa, and al-Khwarizmi.

Yahya was from Tabaristan, the pro-Zoroastrian region south of the Caspian Sea. He was an astrologer, though he is better known for his work in astronomy. [29]

It is not surprising that in addition to their work in mathematics, astronomy and mechanics, the Banu Musa also produced astrological books. [30] Their father, Musa ibn Shakir, was an astrologer, and their mathematics teacher was the same Yahya just mentioned.

Al-Khwarizmi’s reputation as an astronomer was such that his entry
in
*The
Fihrist* notes nothing else: “He was one of the masters of the
science of the stars. Both before and after [confirmation by]
observation,
people relied upon his first and second astronomical tables known as
the
*Sindhind*.
Among his books were: Astronomical tables in two editions, the
first
and the second; The sundial; Operations with the Astrolabe;
Making the Astrolabe; History.” [31]

The *History* is anything but out of place on this list.
Toomer suggests that “in it al-Khwarizmi exhibited an interest in
interpreting
history as fulfilling the principles of astrology.” [32]
Jules Vernet agrees, in his article in the
*Encyclopedia of Islam*,
proposing that the *History* was probably like the astrological
works
of Abu Ma‘shar, al-Nawbakhti and the Andalusian writer Ibn al-Khayyat
(d.
447/1055). [33]
Al-Khwarizmi’s
*History*
is lost, but it seems to have been just the kind of astrological
chronicle
which was produced in the imperial Sasanian library.

It does not seem, then, that it is necessary to assume that the House of Wisdom took a new direction in the reign of al-Ma’mun, though it is certainly possible that it did. We just do not have enough information.

I am not suggesting that astrology and astrological history were
paramount
in the minds of 9th century thinkers. It just appears that the
role
of astrology was a major one within the early ‘Abbasid court, and in
particular
at the *bait al-hikma*, a library which in itself was nothing
like
the great academy of science it has been portrayed as having
been.
The great reputation of the House of Wisdom came later, after the
activities
of the scholars in al-Ma’mun court had become legendary. The *bait
al-hikma* became the institution on which was hung all the
acheivements
of the age by later medieval and modern historians.

**The world of mathematicians**

The definition of what constitutes ‘mathematics’ was different for
medieval
Islamic scientists than it is for us. While we break down
mathematics
into subfields like algebra, topology, and analysis, scholars in the
medieval
Islamic world divided the field, roughly speaking, into arithmetic
(including
algebra), geometry, astronomy, astrology, geography, music, and
sometimes
engineering. This is just an extension of the Greek *quadrivium*
described above. So when considering ‘mathematician’ as a
profession,
it is best to keep in mind that such a practitioner is likely to have
worked
in several of these sub-fields.

Just how did mathematicians in ‘Abbasid times make their livings? Not in higher education—it was reserved primarily for the teaching of the traditional Islamic sciences. You won’t find any mathematician whose main job was professor in one of the pre-madrasa institutions of the ninth century. This continued to be the case through most of the middle ages. But do not take this to mean that Islamic society in general was indifferent of, or even hostile to foreign learning. It is just that such learning was not a part of formal education.

Mathematicians, like philosophers and poets, were hired in the service of the court. Throughout the middle ages patrons included not only the ruler, but other important people as well—usually relatives and advisers of the ruler. Up to the middle of the ninth century the court was only one: that of the caliph in Baghdad. Later, after the power of the caliphate had declined and the empire split into various states, mathematicians were patronized by the court of whatever ruler or important public official found such company useful or desireable.

**Work for hire**

Of course mathematicians were not hired just to sit around and debate the nature of Euclid’s postulates. They served the practical needs of their patrons, whether it was designing canals, casting horoscopes, or devising new and easier methods for dividing estates. The most common specific jobs for our period fall in the subcategories of astrology and astronomy. Astronomy not only served the needs of astrology, but also of religion. The determination of the times of prayer, of the direction to Mecca, and other religious problems require quite sophisticated knowledge of astronomy and spherical trigonometry.

In geography, which is linked to astronomy, one of the big jobs was to calculate the latitude and longitude of major Muslim cities and other locations. As in astronomy, a need was felt to verify and correct the information in Ptolemy’s books.

Three major astronomical/geographical undertakings sponsored by al-Ma’mun were

•the construction of observatories in Baghdad and
Damascus, and the observations made there,

•the calculation of the circumfrence of the earth
based on the measurement of one terrestial degree,

•the construction of a world map.

In mathematics proper, algebra was invented to facilitate the calculations for the distribution of estates. The promulgation of Hindu numerals, together with algorithms for the basic operations, also had practical consequences. Al-Khwarizmi wrote the earliest known treatises on both subjects.

Mathematicians were employed by the state in engineering tasks, too. For example, two of the Banu Musa, Muhammad and Ahmad, were put in charge with eighteen others of a canal project at al-Ja‘fariyya. [34] Ahmad in particular also worked in mechanics. [35] Sind ibn ‘Ali, another scholar in al-Ma’mun’s court, was a well known engineer as well as mathematician and astronomer. [36]

Many of these scientists acted as advisers to the caliph, a role which may be an extension of their jobs as court astrologers. The Banu Musa were employed as political advisers, judging by their activities in the caliphal court. [37] Al-Kindi, the famous Arab philosopher and mathematician, was tutor to Ahmad, son of the caliph al-Mu‘tasim (reigned 833-842). This appointment may have been the cause of a palace dispute between al-Kindi and the Banu Musa.

There was a lot of money to be had in translating works from Greek or Syriac into Arabic, too. Not only did the caliph himself sponsor such works, but as is well known, the Banu Musa themselves paid large sums to translators. These included Hunayn ibn Ishaq (medicine), Thabit ibn Qurra, son of Hunayn (mathematics, etc.), Hubaysh, nephew of Hunayn, and ‘Isa ibn-Yahya. [38]

There was also communication between mathematicians and
artisans.
Though a century later than our period, the work of Abu’l-Wafa’ is
probably
not out of place here. Not only did he write a book titled *On
the Geometric Constructions Necessary for the Artisan*, but he took
part in meetings with the workers. “At some sessions,
mathematicians
gave instructions on certain principles and practices of
geometry.
At others, they worked on geometric constructions of two- or
three-dimenstional
ornamental patterns or gave advice on the application of geometry to
architectural
construction.” [39]

**Work for its own sake**

Theoretical work was also done, even if it may not have been funded directly. Al-Jawhari, who flourished in the first half of the ninth century, attempted to rectify the problem of Euclid’s fifth postulate by proving an equivalent statement. [40] Al-Jawhari was not an odd man at court, either. Al-Nadim reports that he also worked in astronomy and astrology. [41] In astronomy he participated in the observations done at Damascus and Baghdad mentioned above, and he was also in charge of the construction of astronomical instruments. [42]

Several other mathematicians also produced theoretical commentaries
on the *Elements*, including the Banu Musa, al-Kindi, al-Mahani,
Abu
Muhammad al-Hasan, Thabit ibn Qurra, Ishaq ibn Hunain, al-Nairizi, and
Qusta ibn Luqa. [43]
Mathematicians
did equally “useless” work in their studies of Archimedes and others.

And one should not assume that the caliphs and their viziers had only political motives for supporting intellectual activity. It is plausible that many had a real interest in the scholarship produced by their courtiers. In addition we should not overlook the presige gained by a court which produces great mathematics as well as great poetry.

The translation and study of Greek and other foreign scientific works served several purposes: initially, it was used as part of ‘Abbasid propoganda, coupled with the utility of the works. At the same time, as the books of Euclid, Ptolemy, Nicomachus and others were studied, they became the foundation for theoretical research. Politics primed the pump, and research traditions were born.

**Back to algebra**

Let me read again the statement al-Khwarizmi makes in the
introduction
to his *Algebra*. He writes that the caliph al-Ma’mun
himself
“has encouraged me to compose a short work on algebra, confining it to
the fine and important parts of its calculations, such as people
constantly
require in cases of inheritance, legacies, partition, law-suits, and
trade,
and in all their dealings with one another, or where surveying, the
digging
of canals, geometrical computation, and other objects of various sorts
and kinds are concerned.” [44]

As we saw at the beginning of this talk, this is in contrast to our
author’s inclusion of solutions and proofs for the six types of
quadratic
equations—no non-negative coefficients were allowed—which served no
practical
use whatsoever. On the theoretical side, al-Khwarizmi’s *Algebra*,
and books like it from his time, were the seminal works of pure
algebra,
a science which would in the ensuing centuries be extended by Islamic
mathematicians
to polynomials of arbitrary degree, to include irrational coefficients,
and to the solution of cubic equations by intersecting conic
sections. [45]
On the practical side, the *Algebra* stood at the beginning of a
long
tradition of handbooks for the solution of inheritance problems, which
contain nothing about quadratic polynomials.

Eight other mathematicians are known to have produced a book on
algebra
in the ninth century, but of these only Abu Kamil’s survives in
complete
form, and a portion of Ibn Turk’s
*Algebra* has come down to us.
The latter, consisting of the chapter “Logical Necessities in Mixed
Equations”,
dates from the first half of the ninth century. It concerns
solutions
of quadratic equations and has nothing on applications, so we can glean
nothing from it for our present purpose, except to note that the
author,
like al-Khwarizmi, had at least an inclination for pure
mathematics. [46]

The split between the practical and theoretical aspects of algebra
was
made no later than the end of the ninth century. Al-Dinawari (d.
895) produced separate works on algebra and inheritance. [47]
While one might raise the possibility that his lost *Algebra*
might
have served as a practical handbook, the same cannot be said of Abu
Kamil’s
extant book on algebra, written probably no later than about 930.
It follows the lead of al-Khwarizmi’s, and contains many
improvements. [48]
Again the six types of quadratic equation are considered, with proofs
of
each solution.

Abu Kamil also includes 69 worked-out examples—more than al-Khwarizmi’s 40. Over 70% of the book is devoted to such problems. But of these 69 problems, 62 are purely algebraic, most involving the square of the unknown, and many involving the root of a non-square integer. These are problems for the practice of algebra itself. Of the remaining seven problems, (#10-#16), six are concerned with the equal division of a number (money?) among men, and one deals with the purchase of a garment. These are really pure algebra problems stated in real-life terms.

If there are no true applications in Abu Kamil’s *Algebra*,
there
certainly were in his lost book *Inheritance Distribution with the
Aid
of Algebra*.[49] So
he divided
the theoretical from the practical aspects of al-Khwarizmi’s subject
matter.

So you want an answer to the question posed by the title of this talk. The answer is “Yes—al-Khwarizmi was an applied algebraist.” For what it is worth, I imagine that while writing this practical algebra for the caliph, al-Khwarizmi could not resist the temptation to expand the theoretical side as well.

Today there are many mathematicians who share the classical Greek disdain for applications—maybe some of your friends are among them! Some people are often proud that there are no uses for their esoteric branch of homology theory. Being isolated within the mathematics department, they are in professional contact with some analysts and algebraists, but certainly not with engineers or physicians. In the Islamic world the mathematician was likely himself to be also a doctor, an astronomer, and a geographer. Applied mathematics was just part of the job, and the benefits of rigorous proof and practical use were each appreciated.

Notes.

[1] Rosen, 3, adapted in Gutas,
113.

[2] *Republic*, 527b.

[3] *Republic*, 527c.
See also Hoyrup.

[4] There is new evidence, for
example,
that Euclid was not the Platonist we once thought he was. See
Russo.

[5] Thomas II, 615.

[6] though some, such as the
Brethren
of Putiry in the tenth century, were heavily influenced by Neoplatonic
ideas.

[7] Hoyrup, 4.

[8] Gutas, 112. It was also
later found very useful in education, as the Greeks had known.

[9] Jolivet; Rosenthal, 54ff.

[10] Gutas, 23; Al-Nadim, 583.

[11] Yarshater, 54ff, 64-5.

[12] Gutas, 28ff; Pingree (1968),
7ff.

[13] Pingree (1968), 13ff.

[14] Pingree (1968), 11.

[15] Gutas, chapter 5.

[16] Makdisi, 26.

[17] Makdisi, 27.

[18] Gutas, 54.

[19] Gutas, 57.

[20] Gutas, 57.

[21] Gutas, 59.

[22] Saliba, 67.

[23] Gutas, 33.

[24] EI VII, 557.

[25] I do not mean to claim that
these three fields were disjoint from other fields of study such as
philosophy,
theology, or meteorology. The scope of this talk is mathematics,
and my goal here is only to establish the link between mathematics and
astrology.

[26] DSB, 1231.

[27] Gutas, 58.

[28] Gutas; EI; Pingree (1990),
293.

[29] GAS VII, 116.

[30] GAS VII, 129-130.

[31] Al-Nadim, 652.

[32] DSB, 1250.

[33] EI IV, 1071.

[34] EI VII, 640.

[35] EI VII, 640ff.

[36] GAS V, 242.

[37] EI VII, 640ff.

[38] Gutas, 133-4.

[39] Ozdural, 172.

[40] Jaouiche, 37-44.

[41] Al-Nadim, 647.

[42] DSB, 1154-6.

[43] GAS V.

[44] Rosen, 3, adapted in Gutas,
113.

[45] See Rashed.

[46] Sayili.

[47] Al-Nadim, 172.

[48] Levey, 13.

[49] Levey, 7; Al-Nadim, 665.

**Bibliography—standard reference works**

DSB. *Dictionary of Scientific Biography*, C. C.
Gillispie,
ed. New York: Scribner’s, 1970-1990. I used the four
volume *Biographical Dictionary of Mathematicians: Reference
Biographies
from the Dictionary of Scientific Biography*. New York:
Scribner, 1991.

EI. *Encyclopedia of Islam*, 2nd ed. Leiden:
E. J. Brill, 1960-.

GAS V. Fuat Sezgin, *Geschichte des Arabischen Schrifttums,
Band V: Mathematik Bis ca. 430H*. Leiden: E. J.
Brill,
1974.

GAS VI. Fuat Sezgin, *Geschichte des Arabischen
Schrifttums,
Band VI: Astronomie Bis ca. 430H*. Leiden: E. J.
Brill,
1978.

GAS VII. Fuat Sezgin, *Geschichte des Arabischen
Schrifttums,
Band VII: Astrologie - Meteorologie*

*und Verwandtes Bis ca. 430 H*. Leiden: E. J. Brill,
1979.

**Bibliography—books and articles**

Dimitri Gutas, *Greek Thought, Arabic Culture: The
Graeco-Arabic
Translation Movement in Baghdad and Early ‘Abbasid Society (2nd -
4th/8th
- 10th centuries*. London: Routledge, 1998.

Jens Hoyrup, *Integration/Non-integration of Theory and Practice
in
Ancient, Islamic and Medieval Latin contexts*. Preprint 80,
International
Workshop: Experience and Knowledge Structures in Arabic and Latin
Sciences.
Max Plank Institut Fuer Wissenschaftsgeschichte, 1997.

K. Jaouiche, La The'orie des Paralle`s en Pays d’Islam: Contribution a` la Pre'histoire des Ge'ome'tries Non-Euclidiennes. Paris: Vrin, 1986.

Jean Jolivet, “Classifications of the Sciences”, pp. 1008-1025 in
Roshdi
Rashed, ed., *Encyclopedia of the History of Arabic Science* (3
vols.),
London: Routledge, 1996.

Abu Kamil (Martin Levey, tr. & ed.), *The Algebra of Abu
Kamil:*
Kitab fi al-jabr wa’l-muqabala *in a Commentary by Mordecai Finzi*.*
Hebrew text, translation and commentary with special reference to the
Arabic
text*, Madison: University of Wisconsin, 1966.

Al-Khwarizmi, *The Algebra of Mohammed ben Musa*
translated
and edited by Frederic Rosen. Hildesheom: Georg Olms Verlag
(originally
published 1831), 1986.

George Makdisi, *The Rise of Colleges: Institutions of
Learning
in Islam and the West*. Edinburgh: Edinburgh University
Press, 1981.

Al-Nadim, *The Fihrist of al-Nadim: A Tenth-Century Survey
of
Muslim Culture*, edited and translated by Bayard Dodge in two
volumes.
New York: Columbia University Press, 1970.

Alpay Ozdural, ”Mathematicians and Arts: Connections between
Theory
and Practice in the Medieval Islamic

World.” *Historia Mathematica* **27** (2000), 171-201.

David Pingree, *The Thousands of Abu Ma‘shar*.
London:
The Warburg Institute, 1968.

David Pingree, “Astrology.” Chapter 16 (pp. 290-300) in Young,
Latham, Serjeant, eds., *Religion, Learning and Science in the
‘Abbasid
Period*. Cambridge: Cambridge University Press, 1990.

Plato, *The Republic of Plato*, translated with notes and an
interpretive
essay by Allan Bloom. New York: Basic Books, 1968.

Roshdi Rashed, *The Development of Arabic Mathematics:
Between
Arithmetic andAlgebra* Translated by A. F. W. Armstrong.
Dordrecht:
Kluwer, 1994.

Lucio Russo, “The Definitions of Fundamental Geometric Entities
Contained
in Book I of Euclid’s *Elements*.” *Archive for History
of
Exact Sciences* **52** (1998), 195-219.

George Saliba, *A History of Arabic Astronomy: Planetary
Theories
During the Golden Age of Islam*. New York: New York
University,
1994.

Aydin Sayili, *Logical Necessities in Mixed Equations by ‘Abd al
Hamid
Ibn Turk and the Algebra of his Time*. Ankara: Tuerk
Tarih
Kurumu Basimevi, 1962.

Ivor Thomas, tr., *Selections Illustrating the History of Greek
Mathematics*,
in two volumes: Volume I: *From Thales to Euclid*, and
Volume
II: *Aristarchus toPappus*. Harvard:
Heinemann.
Volume I 1929, revised 1991; Volume II 1941, revised 1993.

Ehsan Yarshater, “The Persian Presence in the Islamic World.”
pp. 4-125 in Richard G. Hovannisian and

Georges Sabagh, eds. *The Persian Presence in the Islamic World*.
Cambridge: Cambridge University

Press, 1998.

Jeff Oaks, Department of Mathematics and Computer Science,
University
of Indianapolis. 1400 E. Hanna Ave., Indianapolis, IN 46237.

Send complaints, invective, and refund requests to:
oaks@uindy.edu.