IBN AL-HAWWAM (d.724/1324), HIS WORKS AND THE SECTION ON INSOLUBLE PROBLEMS IN AL-FAWAID AL-BAHAIYYA FI AL-QAWAID AL-HISABIYYA
Imaduddin (or Cemaluddin) Abu Ali Abdullah b. Muhammed al-Hawwam b. Abdurrazzaq al-Harbuwi al-Bagdadi al-Iraqi al-Harezmi al-Şafii, one of the VHI/XIVth century Muslim scientists, was born in Zilkade 643/March 1245 possibly in Baghdad and died in the same city in 724/1323-1324.
After completing his primary education, he learned the rational sciences, most probably in Baghdad, from Nasiruddin al-Tusi, the founder-member of Maragha mathematical-astronomical school. His relation to Maragha school was only confined to his being a student of Nasiruddin al-Tusi and he led an intellectual life independent of the Maragha school.
He has three books on mathematics and four books on tefsir, tasavvuf, ethics and medicine respectively. Ibn al-Hawwam also has "Fawaid" in mathematics which have come down to our age in different ways. Due to his works, he was one of the most prominent figures of his time both in rational and traditional sciences. One of his mathematics books is al-Fawaid al-Bahdiyya fi al-Qawaid al- Hisabiyya which was written in Tsfahan in Şaban 675/January 1276 and was presented to Bahaaddin Muhammed al-Cuvayni. Another book on mathematics by al-Hawwam, al-Risala al-Şamsiyya fi al-Qawaid al-Hisabiyya, is a version of al-Bahaiyya and was probably prepared by Ibn al-Hawwam himself. The basic difference between these two works is that al- Şamsiyya is much more concise and practically oriented. His third book on mathematics is a commentary on the Book X of Euclides's Elements which deals with the geometrical study of irrational numbers.
The last chapter of al-Bahaiyya (Hatime) is studied in detail in this article. Before summarizing this study, it will be appropriate to introduce Ibn al-Hawwam's mathematical works.
Al-Bahaiyya, can be placed within the Pythagorean tradition, a discipline hiiNftl on number mysticism. In Islamic mathematics, number mysticism started after die tntnmliitlon of Nii-onuu hos's Imiodiu no Aritmctica by Sabit b. Kurra into Arabic and was developed by Ihvan al-Safa. The mystic tradition was similar to "Theologoumenates Aritmetikes" in content. al-Bahaiyya can be seen as a follower of this mystic school in Islamic mathematics. In this work, Ibn al-Hawwam takes only hisajb^al-hevai as a subject and does not include hisab al-hindi. Therefore, withuvfne two main hisab traditions of Islamic mathematics, aside from hisab al-muneccimin used by the astronomers, al-Bahaiyya follows the first one.
The algebra explained in al-Bahaiyya can be evaluated to be within the school of analytic algebra based on arithmetic established by al-Kereci. The aim of this school is to express an algebraic formulation with an analytic explanation, without basing it on the geometric ones for arithmetizing the algebra. As it is known, the geometrical and analytical approach to algebra present in Mesopotamia and Ancient Greece was synthetized in Islamic mathematics by Harezmi and Ebu Kamil: they used both methods in solving algebraic equations. Starting with al-Kereci, algebra began to be differentiated from geometry. Samawel continued this practise and finally algebra was competely arithmetized. Algebra of the fourth book and algebraic problems that Ibn al-Havvam solves in the fifth book of al-Bahaiyya totally belong to the analytic approach in algebra.
Al-Bahaiyya was prepared for advanced readers in the form of a "collection of mathematical principles" and displays the level of Islamic mathematics on hisab al-hevai, hisab al-muneccimin, rules of ratio, ilm al-misaha and ilm al-cabr wa al-muqabala. The work is of medium size and level, and recapitulates the mathematical knowledge of its time. For the history of mathematics in general and the history of Islamic mathematics in particular, the thirty three "insoluble problems" that Ibn al-Hawwam noted in the book are the most original aspects of this work. These problems present in the "Hatime" of al-Bahaiyye are studied in this paper.
Some problems of this kind, that are among the intederminate equations, had been dealt with by mathematicians like Diophantus, Ebu Kamil, Ebu'1-Wefa al-Buzcani, al-Kereci, al-Hazin, Samawel, al-Hucendi and Izzuddin al-Zencani before Ibn al-Hawwam. After Ibn al-Hawwam, especially .Cemşid al-Kaşi was interested in such equations in Islamic mathematics. Bahaeddin al-Amili, in his Hulasat al-Hisab, cited only seven of the Ibn al-Hawwam's thirty three problems under the heading of "insoluble problems". The most noteworthy of Ibn al-Hawwam's thirty three problems are the third and twenty, fourth problems which deal with the particular case for n=3 and n=4 of the Fermat's last theorem which shows the impossibility of x n + y n = z n, n>2 equation.
Al-Bahaiyya and its version al-Şamsiyya served as the two main text books in the mathematic al education and studies in Baghdad and Ispahan. Ibn al-Hawwam taught mathematics from these books. This educational activity led to the formation of a bright mathematician and physician called Kemaleddin al-Farisi (d.7 18/1319) who learned the al-Bahaiyya directly from Ibn al-Hawwam in Ispahan.
Al-Bahaiyya was commented on twice by two Muslim mathematicians in the VHIth and XlXth centuries: Esas al-Qawaid fi al-Usul al-Fawaid by Ka-malad-din al-Farisi and hah al-Maqasid fi al-Fara id al-Fawaid by Imaduddin al-Kaşi (d.after 745/1344). Although Imaduddin al-Kaşi stated in his commentary that he would prepare a separate treatise on the insoluble problems of al-Bahaiyye, no such treatise was found. Imaduddin al-Kaşi also wrote some independent commentaries on the some mathematical rules given in al-Bahaiyya.
Al-Bahaiyya and the two commentaries greatly influenced the later Islamic and Ottoman mathematical works on the teaching of mathematics. Ali al- Garbi (VHI/XIVth century) in his Kitab al Mucizat al Necibiyye fi Şerh al- Risalet al-Alaiyye, quoted from al-Bahaiyye and Kemaleddin al-Farisi's commentary; Cemşid al-Kaşi, in his well known work Miftah al-Hisab, took some of the algebraic problems directly from al-Bahaiyye. Molla Lutfi (d.900/1490), in his treatise called Risale fi Tarif al- Hikme, Cabizade Halil Faiz (d. 1124/1712) in his translation al-Savlet al-Hizebriyye fi al-Mesail al-Cebriyye and finally Kuyucaklızade Mehmet Atıf Efendi (d. 1263/1847) in Nihayet al-Elbab fi Tercumet Hulasat al-Hisab, all quoted from the commentary of Kemaleddin al-Farisi. Katip Çelebi (d. 1067/1657) in his uncompleted commentary titled Ahsen el-Hediyye hi Şerh el-Risalet el-Bahaiyyc, compares briefly the al-Bahaiyya and al-Muhammadiyya. Taşköprülüzade (d. 968/1561) in his Miftah al Sa'ade ve Misbah al-Siyade, where he listed the books used by Ottoman scholars, mentioned in the section "hisab al-hevai" al-Farisi's commentary on al-Bahaiyya as the fifth book.
As the commentary of Kemaleddin al-Farisi and lrsad al-Tullab ila İlm al llisah (anonymous, reign of Sultan Beyazid the Second) shows, some of the problems of Ibn al Havvam which were not present either in al-Bahaiyya or al- Ştimslyya were known by later mathematicians. Since there are no other mathematical works by Ibn al-Havvam except the three books that have been mentioned, we may deduce the idea that his other problems have been transmitted to later generations through "Fawaid" tradition.
Lastly, we wish to point out that Salih Zeki (d. 1921), an Ottoman mathematician and historian of mathematics, was the first to emphasize the importance of both Ibn al-Havvam's al-Bahaiyya and the two commentaries. He was also the first to point out the significance of these works within the history of Islamic mathematics and presented the first evaluation of Ibn al-Hawwam's works in his book Asar- 1 Bakiyye .