IBN ALHAWWAM (d.724/1324), HIS WORKS AND THE SECTION ON INSOLUBLE PROBLEMS IN ALFAWAID ALBAHAIYYA FI ALQAWAID ALHISABIYYA
İhsan Fazlıoğlu
Imaduddin (or Cemaluddin) Abu Ali Abdullah b. Muhammed alHawwam b. Abdurrazzaq alHarbuwi alBagdadi alIraqi alHarezmi alŞafii, 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 724/13231324.
After completing his primary education, he learned the rational sciences, most probably in Baghdad, from Nasiruddin alTusi, the foundermember of Maragha mathematicalastronomical school. His relation to Maragha school was only confined to his being a student of Nasiruddin alTusi 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 alHawwam 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 alFawaid alBahdiyya fi alQawaid al Hisabiyya which was written in Tsfahan in Şaban 675/January 1276 and was presented to Bahaaddin Muhammed alCuvayni. Another book on mathematics by alHawwam, alRisala alŞamsiyya fi alQawaid alHisabiyya, is a version of alBahaiyya and was probably prepared by Ibn alHawwam 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 alBahaiyya (Hatime) is studied in detail in this article. Before summarizing this study, it will be appropriate to introduce Ibn alHawwam's mathematical works.
AlBahaiyya, can be placed within the Pythagorean tradition, a discipline hiiNftl on number mysticism. In Islamic mathematics, number mysticism started after die tntnmliitlon of Niionuu hos's Imiodiu no Aritmctica by Sabit b. Kurra into Arabic and was developed by Ihvan alSafa. The mystic tradition was similar to "Theologoumenates Aritmetikes" in content. alBahaiyya can be seen as a follower of this mystic school in Islamic mathematics. In this work, Ibn alHawwam takes only hisajb^alhevai as a subject and does not include hisab alhindi. Therefore, withuvfne two main hisab traditions of Islamic mathematics, aside from hisab almuneccimin used by the astronomers, alBahaiyya follows the first one.
The algebra explained in alBahaiyya can be evaluated to be within the school of analytic algebra based on arithmetic established by alKereci. 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 alKereci, 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 alHavvam solves in the fifth book of alBahaiyya totally belong to the analytic approach in algebra.
AlBahaiyya was prepared for advanced readers in the form of a "collection of mathematical principles" and displays the level of Islamic mathematics on hisab alhevai, hisab almuneccimin, rules of ratio, ilm almisaha and ilm alcabr wa almuqabala. 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 alHawwam noted in the book are the most original aspects of this work. These problems present in the "Hatime" of alBahaiyye 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'1Wefa alBuzcani, alKereci, alHazin, Samawel, alHucendi and Izzuddin alZencani before Ibn alHawwam. After Ibn alHawwam, especially .Cemşid alKaşi was interested in such equations in Islamic mathematics. Bahaeddin alAmili, in his Hulasat alHisab, cited only seven of the Ibn alHawwam's thirty three problems under the heading of "insoluble problems". The most noteworthy of Ibn alHawwam'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.
AlBahaiyya and its version alŞamsiyya served as the two main text books in the mathematic al education and studies in Baghdad and Ispahan. Ibn alHawwam taught mathematics from these books. This educational activity led to the formation of a bright mathematician and physician called Kemaleddin alFarisi (d.7 18/1319) who learned the alBahaiyya directly from Ibn alHawwam in Ispahan.
AlBahaiyya was commented on twice by two Muslim mathematicians in the VHIth and XlXth centuries: Esas alQawaid fi alUsul alFawaid by Kamaladdin alFarisi and hah alMaqasid fi alFara id alFawaid by Imaduddin alKaşi (d.after 745/1344). Although Imaduddin alKaşi stated in his commentary that he would prepare a separate treatise on the insoluble problems of alBahaiyye, no such treatise was found. Imaduddin alKaşi also wrote some independent commentaries on the some mathematical rules given in alBahaiyya.
AlBahaiyya 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 alAlaiyye, quoted from alBahaiyye and Kemaleddin alFarisi's commentary; Cemşid alKaşi, in his well known work Miftah alHisab, took some of the algebraic problems directly from alBahaiyye. Molla Lutfi (d.900/1490), in his treatise called Risale fi Tarif al Hikme, Cabizade Halil Faiz (d. 1124/1712) in his translation alSavlet alHizebriyye fi alMesail alCebriyye and finally Kuyucaklızade Mehmet Atıf Efendi (d. 1263/1847) in Nihayet alElbab fi Tercumet Hulasat alHisab, all quoted from the commentary of Kemaleddin alFarisi. Katip Çelebi (d. 1067/1657) in his uncompleted commentary titled Ahsen elHediyye hi Şerh elRisalet elBahaiyyc, compares briefly the alBahaiyya and alMuhammadiyya. Taşköprülüzade (d. 968/1561) in his Miftah al Sa'ade ve Misbah alSiyade, where he listed the books used by Ottoman scholars, mentioned in the section "hisab alhevai" alFarisi's commentary on alBahaiyya as the fifth book.
As the commentary of Kemaleddin alFarisi and lrsad alTullab 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 alBahaiyya or al Ştimslyya were known by later mathematicians. Since there are no other mathematical works by Ibn alHavvam 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 alHavvam's alBahaiyya 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 alHawwam's works in his book Asar 1 Bakiyye .
