Abu Ja'far Muhammad ibn al-Husayn al- Khazin al-Kurasani

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Education:
Abu Ja’far Muhammad ibn al-Husayn al-Khāzin was a mathematician and
astronomer. Until 1978 it was believed that Abū Ja’far al-Khazin and Abū
Ja’far Muhammad ibn al-Husayn were two different mathematicians. However,
Adil Anbouba showed that they are the same person (see Anbouba,
1978).
Professional Career:
According to the Islamic Encyclopedia, Khazin served as a treasurer during
the reign of the Samanid Prince Mansur ibn Nūh I (961-976) and then as
a Vizier in Nishapur 2 . He came close to the Buwayhid Prince 3 Rukn ad-
Dawlah and his erudite Vizier al-’Amid and spent a part of his life in Rayy.
During peace negotiations between the Buwayhids and the Samanids, al-
Khazin acted as the envoy of the Buwayhids. Al-Khazin later developed
close friendship with the Vizier, al-’Amid, and spent the rest of his life in
research and authoring books under his protection.
Abū Ja’far al-Khazin realized that a cubic equation could be solved geometrically
by means of conic sections. Al-Māhān (ca. 850 AC) had shown that
an unsolved problem in Archimedes’ On the Sphere and Cylinder 4 could be
reduced to the cubic equation:
x 3 + C = ax 2 .
Abū Ja’far based on a commentary to Archimedes’ work by Eutokius of
Ascalon 5 (fifth century AC) and realized that the equation x 3 + C = ax 2
could also be solved by means of conic sections. He also studied the equation
x 3 + y 3 = z 3
and stated (without proof) that it has no solution in positive integers.
Moreover, he also worked on the isoperimetric problem 6 (see Lorch, 1986),
and he wrote a commentary to Book X of Euclid’s Elements 7 .
2 Nishapur or Nishabur is a city in the Khorasan Province in northeastern Iran.
3 The Buyid dynasty controlled most of modern-day Iran and Iraq in the 10th and 11th
centuries.
4 On the Sphere and Cylinder (about 225 BC) was written by Archimedes in two volumes.
Proposition 4 of Book II deals with the problem how to find the surface area of a
sphere and the volume of the contained ball and the analogous values for a cylinder.
5 Eutokius of Ascalon (c. 480 – c. 540) was a Greek mathematician who wrote commentaries
on several Archimedean treatises and on the Apollonian Conics.
6 The isoperimetric problem can be stated as follows: Among all closed curves in the
plane of fixed perimeter, which curve (if any) maximizes the area of its enclosed region?.
7 Euclid’s Elements consist of 13 books written by the Greek mathematician Euclid in