Al-Mu'taman ibn Hud

Abu Amir Yusuf ibn Ahmad ibn Hud, who reigned under the regnal title al-Mu'taman, was one of the most remarkable figures of eleventh-century Al-Andalus: a king who was at the same time a serious and accomplished mathematician. He belonged to the Banu Hud dynasty, the Muslim rulers of the taifa kingdom of Zaragoza in the north-east of the Iberian Peninsula, and he ruled that kingdom from 1081 until his death in 1085. The combination of political power and genuine scientific achievement makes him a rare and fascinating figure in the history of science, for it is uncommon to find a reigning monarch who produced a substantial and original work of pure mathematics.

The eleventh century was the age of the taifa kingdoms in Al-Andalus, the period after the collapse of the unified Umayyad caliphate of Cordoba, when the Muslim west fragmented into numerous competing principalities. Although this fragmentation was a political weakness that would eventually leave Al-Andalus vulnerable to the Christian reconquest, it also produced a remarkable cultural flowering. The rival courts competed not only in arms and diplomacy but also in patronage of poets, philosophers, and scientists. Zaragoza under the Banu Hud became one of the most brilliant of these centres of learning, and its rulers were known as patrons and practitioners of the sciences. Al-Mu'taman was the supreme expression of this tradition: a prince raised in an atmosphere of high culture, who devoted his own energies to mastering the most advanced mathematics of his day.

Al-Mu'taman's great work is a treatise on geometry and number theory titled Kitab al-Istikmal, which may be translated as the Book of Perfection or the Book of Completion. The title reflects its ambition: it was conceived as a comprehensive synthesis that would gather, organise, and complete the mathematical knowledge inherited from the Greeks and from the earlier Islamic mathematicians. In composing the Istikmal, al-Mu'taman drew on a wide range of sources, including the works of Euclid, Archimedes, Apollonius, Ptolemy, the Banu Musa brothers, Thabit ibn Qurra, and especially the great Egyptian mathematician Ibn al-Haytham. Rather than merely copying these authorities, al-Mu'taman selected, rearranged, and supplemented their results, weaving them into a structured whole that covered the foundations of geometry, the properties of numbers, ratios and proportions, conic sections, and other advanced topics.

The most famous single result associated with al-Mu'taman's Istikmal is a geometrical theorem about triangles that, in the modern history of mathematics, is generally known as Ceva's theorem. This theorem concerns the conditions under which three lines drawn from the vertices of a triangle to points on the opposite sides will all pass through a single common point. It is a fundamental result in what is now called the geometry of the triangle. The theorem takes its usual name from the Italian mathematician Giovanni Ceva, who published it in Europe in the year 1678, more than six centuries after al-Mu'taman. Modern scholarship, however, has established that al-Mu'taman stated and proved this same theorem in his Book of Perfection in the eleventh century, long before it was known in Europe. This discovery is a striking example of how the history of mathematics has had to be rewritten as the riches of the Arabic mathematical tradition have become better known, and it gives al-Mu'taman a clear and honourable place among the founders of the geometry of the triangle.

The Istikmal contained many other notable results besides this one. Al-Mu'taman treated problems concerning the properties of conic sections, building on the work of Apollonius and Ibn al-Haytham; he dealt with questions of ratio and proportion in the tradition of Euclid; and he addressed problems in number theory and the theory of magnitudes. The work was designed as a kind of mathematical encyclopaedia, intended to bring the scattered achievements of the Greek and Islamic traditions into a single coherent framework. It is important to recognise that, although al-Mu'taman built upon his predecessors, his selection and arrangement were themselves a creative act, and in a number of places he supplied his own proofs and his own treatments of difficult problems. The breadth of the synthesis and the level of mathematical sophistication it required were considerable, and they testify to the genuine depth of his understanding.

The composition of such a work by a man who was also a ruling prince is in itself extraordinary. Al-Mu'taman appears to have written most or all of the Istikmal before he came to the throne in 1081, during the years when he was still a prince in the service of his father, al-Muqtadir ibn Hud, who was himself a noted patron of learning. When al-Mu'taman became king of Zaragoza, his reign was brief, lasting only about four years until his death in 1085. He was succeeded by his son, al-Musta'in, and the Banu Hud continued to rule in the region for some decades more, though the political fortunes of the taifa kingdoms continued their slow decline in the face of the advancing Christian kingdoms of the north.

The later history of the Istikmal is itself a story of loss and recovery. For a long time the complete text was believed to be lost, and al-Mu'taman was known mainly through references to his work in the writings of other mathematicians, such as the later scholar Maimonides, who praised him highly. It was only in the second half of the twentieth century that manuscripts containing substantial portions of the Istikmal were identified in libraries, allowing modern historians of mathematics to study the work directly and to confirm the remarkable claims about its contents, including the presence of the theorem now named after Ceva. This rediscovery transformed al-Mu'taman from a shadowy name into a securely attested major mathematician of the medieval west.

Al-Mu'taman ibn Hud died in 1085 in Zaragoza, the city he had ruled. He left behind a reputation that has only grown with time, as the true scope of his mathematical achievement has become clear to scholars. The respect in which he was held by later mathematicians is significant: the great Jewish philosopher and scholar Moses Maimonides, writing roughly a century after al-Mu'taman, named him among the most distinguished mathematicians, an indication of the high standing his work enjoyed within the learned circles of the medieval world.

The legacy of al-Mu'taman ibn Hud is many-sided. He stands as one of the very few examples in world history of a ruling monarch who was also a creative mathematician of the first rank, a living embodiment of the union of power and learning that the best traditions of Islamic civilisation held as an ideal. His Book of Perfection represents an ambitious attempt to gather and complete the whole inheritance of Greek and Islamic mathematics, and it preserves and extends the work of earlier masters such as Ibn al-Haytham. Above all, his early statement and proof of the theorem of the triangle that Europe would only rediscover six centuries later stands as a permanent reminder of the depth and originality of the Andalusian mathematical tradition. For the modern student, al-Mu'taman ibn Hud is a powerful symbol of a civilisation in which the pursuit of knowledge was honoured at the very highest levels of society, and in which a king could find no greater glory than to perfect the science of geometry.