Note:  html does not allow me to write Arabic names with the proper diacritical marks.  For example, in the name al-Khwarizmi, there should be a bar above the ‘a’ and the last ‘i’.

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.
 

Was al-Khwarizmi an applied algebraist?

Jeff Oaks, University of Indianapolis


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.

Lets 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.

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