Q?H? (OR AL-K?H?), AB? SAHL WAYJAN IBN RUSTAM AL-

(fl. Baghdad, c. 970–1000),

mathematics, astronomy. For the original article on al-Q?himacr; see DSB, vol. 11.

Ab? Sahl Wayjan, born Rustam al-Q?h?i (in many manuscripts “al-K?h?”), has come to be recognized by modern scholarship as one of the great geometers of tenth-century Islam. He was the only geometer in medieval Islam to obtain exact results on centers of gravity, and he also gave an elegant method for finding the side of a regular heptagon and the volume of a segment of a paraboloid. One of a number of geometers who worked in eastern Iraq and Iran, he enjoyed the patronage of three B?yid rulers: 'Adud al-Daulah, Samsam al-Daulah, and Sharaf al-Daulah, whose combined reigns cover the period 962–989. His contemporaries regarded his work highly, Ibn al-Haytham referring to al-Q?h?i’s On the Measurement of the Paraboloid and al-B?ir?nī citing his On the Complete Compass. In the twelfth century, 'Umar alKhayyami cited him as one of the “distinguished mathematicians of Iraq,” and al-Khazini summarized some of al-Quhi’s work on centers of gravity in the former’s Balance of Wisdom.

Work in Geometry Al-Q?h?’s more than thirty extant treatises reveal him as primarily a geometer, a subject he described in the preface to his treatise on the regular heptagon as “the leader who is to be followed when it comes to honesty.” In his correspondence with Ab? Ishaq alS?b?, he praised mathematics as a demonstrative science, whose goal was to seek the truth—not numerical approximations.

In his treatise On Rising Times, he wrote that he had also investigated astronomy as well as centers of gravity and optics. His Perfect Compass, for example, represented a step beyond Ibn Sin?’s pointwise constructions of conic sections and described an instrument al-Q?h? characterized as useful for drawing these sections on sundials and astrolabes.

Yet these areas appealed to him primarily as sources for geometrical problems. His lengthy Treatise on the Construction of the Astrolabe with Proofs was principally devoted to the problem of completing the lines of an astrolabe, given certain of its circles and points. His On the Distance from the Center of the Earth to the Shooting Stars set out a method that is mathematically correct, though impractical at the time, for finding the distance and size of these objects. In his Rising Times, al-Q?h? took a conservative stance vis-à-vis the new trigonometrical theorems he had heard about, and he showed how the classical Menelaus’s theorem might be used to solve a sequence of standard problems in spherical astronomy. (He emphasized that he had not devoted much attention to studying methods for constructing astronomical tables.)

Al-Q?h? took special interest in problems stemming from the works of Euclid, Apollonius, and Archimedes. In his Revision of Euclid’s Elements, I, he reorganized the latter by eliminating all of its constructions, using the parallel postulate much earlier, devising a new proof of the Pythagorean theorem, and giving an ostensible proof of the fourth postulate on the equality of right angles.

Al-Q?h?s studies of Elements, II provide twelve new propsoitions, very much in the spirit of the first ten propositions of that work, as well as a short Lemmas to the Conics, whose introduction describes it as “necessary in the second and third books of The Conics.”

Archimedean Tradition Unique in medieval Islam are al Q?h?’s results on centers of gravity of plane and solid figures, results very much in the tradition of Archimedes. This research, he said in the preface to On the Volume of the Paraboloid, motivated his work on that question. Although al-Kh?zinī’s Balance of Wisdom summarizes some of his work on centers of gravity, scholars have only al-Q?h?’s correspondence with al-S?b? on the subject, in which he correctly located the centers of gravity of triangles (and cones) and segments of parabolas (and paraboloids), as well as of hemispheres (a result not found in Archimedes’s works). He conjectured, on the basis of these results, that the center of gravity of a semicircle divides the radius perpendicular to its diameter into two parts, so that the part nearer the diameter has to the radius the ratio of 3:7. He was fully aware, and defended the implication, of this result, namely that the ratio of the circumference of a circle to its diameter is 28/9, an insistence that earned him the incredulity of his correspondent and the severe censure of Ab? al-Fut?h al-Sari in his Falsification of the Premises of the Discourse of Ab? Sahl al-Q?h?.

Also closely related to the medieval Islamic tradition of Archimedes’s work is Al-Q?h?’s Construction of a Regular Heptagon in the Circle. By the mid-tenth century, geometers such as al-Sijz? had become dissatisfied with Greek verging constructions, calling them “moving geometry.” (Verging constructions demanded that one insert a line segment of given length so that its endpoints rest on two given curves and so that it points [or “verges”] towards a given point.) Archimedes’s construction of the regular heptagon went beyond the usual verging construction in demanding not that the line inserted between two straight lines have a certain length but that the two triangles created thereby have equal areas. (One of al-Q¯h?’s contemporaries, Ab? al-J?d, described this particularly opaque auxiliary construction as “perhaps more difficult than the task itself.”) It was in the context of this discussion of the limits of a proper construction that al-Quhi wrote, in his preface to the work, that he had done what Archimedes had been unable to do. By this he meant that his construction used not verging but the intersection of conic sections.

Influence of Apollonius Al-Q?himacr;’s On Tangent Circles deals with constructing circles tangent to two given circles or straight lines (or passing through two given points) and having their centers on a given line. This is reminiscent of Apollonius’s famous three-circles problem. Al-Q?h? also considered the case when the line is not just straight or a conic section but any curved line (though what he meant by that is not specified).

Al-Q?h? used freely the classical method of analysis and synthesis, familiar from his study of the works of Apollonius. One example is his Drawing Two Lines from a Known Point, a work probably motivated by his Treatise on the Astrolabe, in which he cited two results from Drawing Two Lines. Among the dozen problems he considers in Drawing Two Lines, the following is a typical one: Point A and line (not necessarily straight) BG are given; assuming this, draw two straight line segments from A to BG, containing a given angle, so that the two segments AB and AG have to each other a given ratio.

Al-Q?h?’s analysis of each problem reduces it to a previously analyzed problem, but no synthesis is ever given. Work like this on analysis was likely the motivation for his treatise, Additions to the Data, which adds a number of new propositions and a new notion to Euclid’s Data.

SUPPLEMENTARY BIBLIOGRAPHY

WORKS BY AL-QUHI

Berggren, J. L. “The Correspondence of Ab? Sahl al-K?h? and Ab? Ishaq al-S?b?: A Translation with Commentaries.” Journal for the History of Arabic Science 7 (1983): 39–124.

Hogendijk, Jan P. “Al-K?h?’s Construction of an Equilateral Pentagon in a Given Square.” Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften 1 (1984): 100-144.

———. “Corrections and Supplements: ‘Al-K?h?’s Construction of an Equilateral Pentagon in a Given Square.’” Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften 4 (1987–1988): 267.

Young, Gregg de. “Ab? Sahl’s Additions to Book II of Euclid’s ‘Elements.’” Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften 7 (1991–1992): 73–135.

Berggren, J. L. “Ab? Sahl al-K?h?’s Treatise on the Construction of the Astrolabe with Proof: Text, Translation, and Commentary.” Physis 31 (1994): 141–252.

Rashed, Roshdi. Les mathématiques infinitésimales du IXe au XIe siècle. Vol. I. Fondateurs et commentateurs: Ban? M?s?, Ibn Qurra, Ibn Sin?n, al-Kh?zin, al-Q?h?, Ibn al-Samh, Ibn H?d. London: Al-Furqan Islamic Heritage Foundation, 1996. Contains Arabic text and translation of both versions of al-K?h?’s treatise on the measurement of the paraboloid.

Berggren, J. L. “Al-K?h?’s ‘Filling a Lacuna in Book II of Archimedes’ in the Version of Nasir al-Din al-Tusi.” Centaurus 38 (1996): 140–207.

Rashed, Roshi. “Al-Q?h? vs. Aristotle: On Motion.” Arabic Sciences and Philosophy 9 (1999): 7–24.

Berggren, J. L., and Glen Van Brummelen. “Ab? Sahl al-K?h? on ‘Two Geometrical Questions.’” Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften 13 (1999–2000): 165–187.

———. “Ab? Sahl al-K?h?’s ‘On the Ratio of the Segments of a Single Line that Falls on Three Lines.’” Suhayl 1 (2000): 11–56.

Rashed, Roshdi. “Al-Q?h?: From Meteorology to Astronomy.”Arabic Sciences and Philosophy 11, no. 2 (2001): 157–204.

Van Brummelen, Glen, and J. L. Berggren. “Abū Sahl al-Kūhī on the Distance to the Shooting Stars.” Journal for the History of Astronomy32, no. 2 (2001): 137–151.

Berggren, J. L., and Glen Van Brummelen. “Ab? Sahl al-K?h? on Drawing Two Lines from a Point with a Known Angle.” Suhayl 2 (2001): 161–198.

———, and Glen Van Brummelen. “Ab? Sahl al-K?h?i on Rising Times.” SCIAMVS 2 (2001): 31–46.

———, and Glen Van Brummelen. “From Euclid to Apollonius: Al-K?h?’s Lemmas to the Conics.” Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften 15 (2002–2003): 165–174.

———, with Jan P. Hogendijk. “The Fragments of Ab? Sahl alK?h?’s Lost Geometrical Works in the Writings of al-Sijz?i.” In Studies in the History of the Exact Sciences in Honour of David Pingree, edited by Charles Burnett, Jan P. Hogendijk, Kim Plofker, et al. Leiden: Brill, 2003.

Abgrall, Philippe. Le développement de la géométrie aux IXe–XIe Siècles: Al-Q?h?. Paris: Albert Blanchard, 2004.

OTHER SOURCES

Berggren, J. L. “Tenth-century Mathematics through the Eyes of Ab? Sahl al- K?h?.” In The Enterprise of Science in Medieval Islam, edited by Jan P. Hogendijk and Abdelhamid I. Sabra. Cambridge, MA, and London, England: MIT Press, 2003.

Hogendijk, Jan P. “Greek and Arabic Constructions of the Regular Heptagon.” Archive for History of Exact Sciences 30 (1984): 197–330.

Sesiano, Jacques. “Note sur trois théorèmes de Mécanique d’al-Q?h? et leur conséquence.” Centaurus 22, no. 4 (1978–1979): 281–297.

Sezgin, Fuat. Geschichte des Arabischen Schrifttums. Vol. 5, Mathematik. Leiden, Netherlands: E. J. Brill, 1974.

J. L. Berggren