(fl. Baghdad, c. 970–1000)
Abū Sahl Wayjan ibn
Rustam al-Qūhī (al-Kūhī; Persian: ابوسهل بیژن کوهی Abusahl Bijan-e Koohi) was a Persian mathematician,
physicist and astronomer. He was from Kuh (or Quh), an area in Tabaristan,
Amol, and flourished in Baghdad in the 10th century. He is considered one of
the greatest Muslim geometers, with many mathematical and astronomical writings
ascribed to him.
Engraving of al-Qūhī's perfect compass to draw conic sections
Al-Qūhī was the leader
of the astronomers working in 988 AD at the observatory built by
the Buwayhid amir 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 an equilateral pentagon into a
square, resulting in a fourth degree equation. He also wrote a
treatise on the "perfect compass", a compass with one leg of variable
length that allows users to draw any conic section: straight
lines, circles, ellipses, parabolas and hyperbolas. It
is likely that al-Qūhī invented the device.
Like Aristotle,
al-Qūhī proposed that the weight of bodies varies with their distance
from the center of the Earth.
The correspondence
between al-Qūhī and Abu Ishaq al-Sabi, a high civil servant interested in
mathematics, has been preserved.
Abū Sahl Wayjan, born
Rustam al-Qūhīi (in many manuscripts “al-Kūhū”), 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 Būyid 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-Qūhīi’s On the
Measurement of the Paraboloid and al-Bīirūnī 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-Qūhī’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 Abū
Ishaq alSābī, he praised mathematics as a demonstrative science, whose goal was
to seek the truth—not numerical approximations.
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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 Sinā’s pointwise constructions of conic sections
and described an instrument al-Qūhī 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-Qūhī
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-Qūhū 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-Qūhīs 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 Qūhī’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-Khāzinī’s Balance of
Wisdom summarizes some of his work on centers of gravity, scholars
have only al-Qūhī’s correspondence with al-Sābī 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 Abū al-Futūh al-Sari in his Falsification of the
Premises of the Discourse of Abū Sahl al-Qūhī.
Also closely related to
the medieval Islamic tradition of Archimedes’s work is Al-Qūhī’s Construction
of a Regular Heptagon in the Circle. By the mid-tenth century, geometers
such as al-Sijzī 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¯hī’s contemporaries, Abū al-Jūd, 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-Qūhimacr;’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-Qūhī 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-Qūhī 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-Qūhī’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.
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