. 940-c. 1000 Abū Sahl Wayjan ibn Rustam al-Qūhī (al-Kūhī

Abū Sahl Wayjan ibn Rustam al-Qūhī (al-Kūhī; (c. 940-c. 1000)                       Persian: ابوسهل بیژن کوهی‎ Abusahl Bijan-e Koohi) was a Persian^[1]mathematician, physicist and astronomer. He was from Kuh (or Quh), an area in Tabaristan, Amol, and flourished in Baghdadin the 10th century. He is considered one of the greatest Muslim geometers, with many mathematical and astronomical writings ascribed to him.

Engraving of al-Quhi's perfect compass to draw conic sections

He was the leader of the astronomers working in 988 AD at the observatory built by theBuwayhid Sharaf al-Dawla in Badhdad. He wrote a treatise on the astrolabe in which he solves a number of difficult geometric problems.

In mathematics he devoted his attention to those Archimedean and Apollonian problems leading to equations higher than the second degree. He solved some of them and discussed the conditions of solvability. For example, he was able to solve the problem of inscribing a regular pentagon into a square, resulting in an equation of fourth degree.^[2]He also wrote a treatise on the "perfect compass", a compass with one leg of variable length that allows to draw any conic section: straight lines, circles, ellipses, parabolasand hyperbolas. It is likely that al-Quhi invented the device.^[3]

Like Aristotle, al-Quhi proposed that the heaviness of bodies vary with their distance from the center of the Earth.^[4]

The correspondence between al-Quhi and Abu Ishaq al-Sabi, a high civil servant interested in mathematics, has been preserved.^[5]

References[edit]

Rashed, Roshdi (1996). Les Mathématiques Infinitésimales du IXe au XIe Siècle 1: Fondateurs et commentateurs: Banū Mūsā, Ibn Qurra, Ibn Sīnān, al-Khāzin, al-Qūhī, Ibn al-Samḥ, Ibn Hūd. London. Reviews: Seyyed Hossein Nasr (1998) in Isis89 (1) pp. 112-113; Charles Burnett (1998) in Bulletin of the School of Oriental and African Studies, University of London61 (2) p. 406.

M. Steinschnieder, Lettere intorno ad Alcuhi a D. Bald. Boncompagni (Roma, 1863)

Suter, Die Mathematiker und Astronomen der Araber (75-76, 1900).

Jan Hogendijk: Two beautiful geometrical theorems by Abu Sahl Kuhi in a 17th century Dutch translation, Ta'rikh-e Elm: Iranian Journal for the History of Science 6 (2008), 1-36

John Lennart Berggren, Hogendijk: The Fragments of Abu Sahl al-Kuhi's Lost Geometrical Works in the Writings of al-Sijzi, in: C. Burnett, J.P. Hogendijk, K. Plofker, M. Yano (eds): Studies in the History of the Exact Sciences in Honour of David Pingree, Leiden: Brill, 2003, pp. 605–665

External links[edit]

Berggren, Len (2007). "Kūhī: Abū Sahl Wījan ibn Rustam [Wustam] al‐Kūhī [al‐Qūhī]". In Thomas Hockey et al. The Biographical Encyclopedia of Astronomers. New York: Springer. p. 659. ISBN 978-0-387-31022-0. (PDF version)

O'Connor, John J.; Robertson, Edmund F., "Abu Sahl Waijan ibn Rustam al-Quhi", MacTutor History of Mathematics archive, University of St Andrews.

Dold-Samplonius, Yvonne (2008) [1970-80]. "Al-Qūhī (or Al-Kūhī), Abū Sahl Wayjan Ibn Rustam". Complete Dictionary of Scientific Biography. Encyclopedia.com.

There are two spellings of al-Quhi's name in English which seem to appear about equally often, namely al-Quhi and al-Kuhi.

We can deduce from al-Quhi's name that he came from the village of Quh in Tabaristan. He was brought up during the period that a new dynasty was being established which would rule over Iran. The Buyid Islamic dynasty ruled in western Iran and Iraq from 945 to 1055 in the period between the Arab and Turkish conquests. The period began in 945 when Ahmad Buyeh occupied the 'Abbasid capital of Baghdad. The high point of the Buyid dynasty was during the reign of 'Adud ad-Dawlah from 949 to 983. He ruled from Baghdad over all southern Iran and most of what is now Iraq. A great patron of science and the arts, 'Adud ad-Dawlah supported a number of mathematicians at his court in Baghdad, including al-Quhi, Abu'l-Wafa and al-Sijzi.

In 969 'Adud ad-Dawlah ordered that observations be made of the winter and summer solstices in Shiraz. These observations of the winter and summer solstices were made by al-Quhi, al-Sijzi and other scientists in Shiraz during 969/970.

Sharaf ad-Dawlah was 'Adud ad-Dawlah's son and he became Caliph in 983. He continued to support mathematics and astronomy so al-Quhi remained at the court in Baghdad working for the new Caliph. Sharaf ad-Dawlah required al-Quhi to make observations of the seven planets and in order to do this al-Quhi had an observatory built in the garden of the palace in Baghdad. The instruments in the observatory were built to al-Quhi's own design and installed once the building was complete. Al-Quhi was made director of the observatory and it was officially opened in June 988.

A number of scientists were present at the opening. One in particular, the famous mathematician and astronomer Abu'l-Wafa, is worthy of mention. He was also employed at the court of Sharaf ad-Dawlah. Another who was present at the opening was Abu Ishaq al-Sabi. Al-Sabi was a high ranking official in Baghdad who was interested in mathematics. We mention later in this article correspondence between al-Quhi and al-Sabi. Some accurate observations were made but the observatory ceased work in 989 on the death of Sharaf ad-Dawlah. The Buyid dynasty was by this stage beginning to lose control of the empire. The economy was on a downward path, and rebellions in the army made the ruler's life difficult. Fine cultural activities such as an observatory took a lower priority.

Our description of al-Quhi's life has highlighted his work in astronomy. However, it is in mathematics that he is more famous, being the leading figure in a revival and continuation of Greek higher geometry in the Islamic world. The geometric problems that al-Quhi studied usually led to quadratic or cubic equations. Nasir al-Din al-Tusi described one of the problems considered by al-Quhi writing (see for example [1]):-

To construct a sphere segment equal in volume to a given sphere segment and equal in area to a second sphere segment - a problem similar to but more difficult than related problems solved byArchimedes - Al-Quhi constructed the two unknown lengths by intersecting an equilateral hyperbola with a parabola and rigorously discussed the conditions under which the problem is soluble.

Al-Quhi's solution to the problem is given in [5]. It is a classical style of solution using results from Euclid's Elements, Apollonius's Conics and Archimedes' On the sphere and cylinder. If a solution exists, al-Quhi showed that it will have coordinates which lie on a particular rectangular hyperbola that he has constructed. Of course, al-Quhi does not express the mathematics in these modern terms but rather in the usual classical geometry of ancient Greek mathematics. Next al-Quhi introduces the "cone of the surface" which, after many deductions, leads to showing that the solution has coordinates lying on a parabola. The problem is then beautifully solved as the intersection of the two curves.

In another treatise On the construction of an equilateral pentagon in a known square al-Quhi solves the problem given in the title again using the intersection of two conic sections, this time two hyperbolas. Although it is impossible to inscribe a regular pentagon in a square, an equilateral pentagon can be inscribed in two ways. One, which requires the solution of a quadratic equation, had been found by Abu Kamil in the ninth century. The other, which requires the solution of a quartic equation, is the one presented by al-Quhi. Details of this treatise are given in [6] (see the corrections and additions of [7]), and [8].

Al-Quhi also described a conic compass, a compass with one leg of variable length, for drawing conic sections in the treatise On the perfect compass [1]:-

... he first described the method of constructing straight lines, circles, and conic sections with this compass, and then treated the theory. He concluded that one could now easily constructastrolabes, sundials and similar instruments.

Indeed al-Quhi did consider the problem of constructing astrolabes in On the construction of the astrolabe. The astrolabe was an instrument used to observe altitudes, and it provided a mechanical means to transform celestial coordinates between an equatorial system and one based on the horizon. This treatise is in two Books, the first being divided into four chapters, the second book into seven chapters.

There are a number of difficult mapping problems solved by al-Quhi in this work. In particular, using a method resembling descriptive geometry, he maps circles on the sphere into the equatorial plane. After manipulation, they are mapped back again onto the sphere in a remarkable piece of visualisation. Despite the appearance of the work being of practical use in constructing an astrolabe, it would appear that al-Quhi was more interested in the mathematics for its own sake than he was in giving a practical manual.

Finally we should mention the correspondence between al-Quhi and al-Sabi which we mentioned above. It is known that there were at least six letters exchanged but only details of four survive. These are given in both Arabic and English in [3]. Topics covered are quite varied, ranging from a discussion of what "known" means to solutions of specific problems such as the following Suppose we are given a circle and two intersecting straight lines l and m. Suppose the tangent to the circle at a point T meets l at L and m at M. How can one choose T so that TL : TM is equal to a given ratio? Perhaps the most interesting parts of the correspondence are six theorems given by al-Quhi concerning the centres of gravity of various figures. Five of the six results are correct but the sixth is false. It states that the centre of gravity of a semicircle divides the radius in the ratio 3 : 7. From this false result al-Quhi deduces the equally false result that π = ²⁸/₉. Even the best mathematicians can make mistakes!

Al-Qūhī’s names indicate his Persian origin: Al-Qūhī means “from Quh,” a village in Tabaristan; and Rustam is the name of a legendary Persian hero. At the peak of his scientific activity he worked in Baghdad under the Buwayhid caliphs ʿẠud al-Dawla and his son and successor Sharaf al-Dawla.

In 969/970 al-Q̅h̄ assisted at the observations of the winter and summer solstices in Shiraz. These observations, ordered by ʿẠud al-Dawla, were directed by Abū’l ̣usayn ʿAbd al Rahmän ibn ʿUmar al-̣ūfī; Ạmad ibn Mụammad ibn ʿAbd al Jal̄al Sijz̄ and other scientists were also present. In 988 Sharaf al-Dawla instructed al-Qūhī to observe the seven planets, and Al-Qūhī constructed a building in the palace garden to house instruments of his own design. The first observation was made in June 988 in the presence of Al-Qūhī, who was director of the observatory; several magistrates (qudät); and the scientists Abu’l Wafä, Ahmad ibn Muhammad al-Säghäni, Abū’l Hasan Muhammad al-Sämarri, Abu’l Hasan al-Maghribi, and Abū Ishäq Ibrähim ibn Hiläl ibn Ibrähim ibn Zahrūn al Ṣäbi. Correspondence between Abū Ishaq and Al-Qūhī still exists. They very accurately observed the entry of the sun into the sign of Cancer and, about three months later, its entry into the sign of Libra. Al-Birūn’i related that activity at al-Qūhi’s observatory ceased with the death of Sharaf al-Dawla in 989.

Al-Qūhī, whom al-Khayyämi considered to be an excellent mathematician, worked chiefly in geometry. In the writings known to us he mainly solved geometrical problems that would have led to equations of higher than the second degree. Naṣīr al Dīn al Tūsi adds to his edition of Archimedes’ Sphere and Cylinder the following note by Al-Qūhī; “To construct a sphere segment equal in volume to a given sphere segment, and equal in surface area to a second sphere segment—a problem similar to but more difficult than related problems solved by Archimedes—Al-Qūhī constructed the two unknown lengths by intersecting an equilateral hyperbola with a parabola and rigorously discussed the conditions under which the problem is solvable.”

The same precision is found in Risäla ft istikhräj dil ʿal-musabbaʿ al-mutasäwil-adläʿ fi’ d-däira(“Construction of the Regular Heptagon”), a construction more complete than the one attributed to Archimedes. Al-Qūhi’s solution is based on finding a triangle with an angle ratio of 1:2:4. He constructed the ratio of the sides by intersecting a parabola and a hyperbola, with all parameters equal. Al-Sijzi, who claimed to follow the method of his contemporary Abū Saʿd al-ʿAlä ibn Sahl, used the same principle. The latter, however, knew al-Qūhi’s work, having written a commentary on the treatise Kitäb Sanʿat al-asturläh (“On the Astrolabe”). Another method used by Al-Qūhī is found in al-Sijzī’s treatise Risala ft qismat atzaiya (“On Trisecting an Angle”).

Again, in Risala ft istikhraj misahat al-mujassam al-mukafi (“Measuring the Parabolic Body”), Al-Qūhī gave a somewhat simpler and clearer solution than Archimedes had done. He said that he knew only Thabit ibn Qurra’s treatise on this subject, and in three propositions showed a shorter and more elegant method. Neither computed the paraboloids originating from the rotation of the parabola around an orDīnate. That was first done by Ibn al-Haytham, who was inspired by Thabifs and al-Quhfs writings. Although he found aJ-Quhi’s treatment incomplete, Ibn a]-Haytham was nevertheless influenced by his trend of thought.

Analyzing the equation x³+ a = cx², Al-Qūhī concluded that it had a (positive) root if a a≤4c³/27. This result, already known to Archimedes, apparently wafl not known to al-Khayyami, whose solution is less accurate. AI-Khayyamf also stated that Al-Qūhī could not solve the equation x³+ 13.5x + 5 = 10x² while Abu’l Jud was able to do so. (Abu’I Jud, a contemporary of al-Bfruni, worked on geometric problems leaDīng to cubic equations; his main work is not extant.)

In connection with Archimedean mathematics, Steinschneider stated that Al-Qūhī also wrote a commentary to Archimedes’ Lemmata. In I. A. Borelli’s seventeenth-century Latin edition of the Lemmata (or Liber assumptorum), there is a reference to Al-Qūhī.

Al-Qūhī was the first to describe the so-called conic compass, a compass with one leg of variable length for drawing conic sections. In this clear and rather general work, Risala fi’l birkar al-tamm (“On the Perfect Compass”) he first described the method of constructing straight lines, circles, and conic sections with this compass, and then treated the theory. He concluded that one could now easily construct astrolabes, sundials, and similar instruments. Al-Biruni asked his teacher Abu Nasr Mansur ibn ’Iraq for a copy of the work; and in al-Biruni, Ibn al-Husayn found a reference to al-Quhfs treatise. Having tried in vain to obtain a copy, Ibn al-Husayn wrote a somewhat inferior work on the subject (H. Suter, Die Mathe-matiker und Astronomen der Araber und ihre Werke [Leipzig, 1900], p. 139).

Al-Qūhī also produced works on astronomy (Brockelmann lists a few without titles), and the treatise on the astrolabe mentioned above. Abu Nasr Mansur ibn clraq, who highly esteemed Al-Qūhī, gave proofs for constructions of azimuth circles by Al-Qūhī in his Risala fI dawa ir as-sumut ft al-asturlcib (“Azimuth Circles on the Astrolabe”).