- GEOMETRY IN THE MUSLIM WORLD 335
FIGURE 9. Thabit ibn-Qurra's Pythagorean theorem.
4.3. Al-Kuhi. A mathematician who devoted himself almost entirely to geometry
was Abu Sahl al-Kuhi (ca. 940-ca. 1000), the author of many works, of which some
30 survive today. Berggren (1989), who has edited these manuscripts, notes that
14 of them deal with problems inspired by the reading of Euclid, Archimedes,
and Apollonius, while 11 others are devoted to problems involving the compass,
spherical trigonometry, and the theory of the astrolabe. Berggren presents as an
example of al-Kuhi's work the angle trisection shown in Fig. 10. In that figure
the angle ö to be trisected is ABG, with the base BG horizontal. The idea of the
trisection is to extend side AB any convenient distance to D. Then, at the midpoint
of BD, draw a new set of mutually perpendicular lines making an angle with the
horizontal equal to ø/2, and draw the rectangular hyperbola through Β having
those lines as asymptotes. Apollonius had shown (Conies, Book 1, Propositions 29
and 30) that D lies on the other branch of the hyperbola. Then BE is drawn equal
to BD, that is a circle through D with center at Β is drawn, and its intersection
with the hyperbola is labeled E. Finally, EZ is drawn parallel to BG. It then
follows that ø = ÄΑÆΕ = IZBE + ÄÆΕΒ = 30, as required.
4.4. Al-Haytham. One of the most prolific and profound of the Muslim math-
ematician-scientists was Abu Ali ibn al-Haytham (965-1040), known in the West
as Alhazen. He was the author of more than 90 books, 55 of which survive.^14 A
significant indication of his mathematical prowess is that he attempted to recon-
struct the lost Book 8 of Apollonius' Conies. His most famous book is his Treatise
on Optics (Kitab al-Manazir) in seven volumes. The fifth volume contains the
problem known as Alhazen's problem: Given the location of a surface, an object,
and an observer, find the point on the surface at which a light ray from the object
will be reflected to the observer. Rashed (1990) points out that burning-mirror
problems of this sort had been studied extensively by Muslim scholars, especially
by Abu Saad ibn Sahl some decades before al-Haytham. More recently (see Guizal
and Dudley, 2002) Rashed has discovered a manuscript in Teheran written by ibn
Sahl containing precisely the law of refraction known in Europe as Snell's law, after
Willebrod Snell (1591-1626) or Descartes' law.^15 The law of refraction as given by
Ptolemy in the form of a table of values of the angle of refraction and the angle of
incidence implied that the angle of refraction was a quadratic function of the angle
of incidence. The actual relation is that the ratio of the sines of the two angles is
a constant for refraction at the interface between two different media.
(^14) Rashed (1989) suggested that these works and the biographical information about al-Haytham
may actually refer to two different people. The opposite view was maintained by Sabra (1998).
(^15) According to Guizal and Dudley, this law was stated by Thomas Harriot (1560-1621) in 1602.