1115–1130 Al-Khazini
, see Abū Ja'far al-Khāzin.
Abu
al-Fath Abd al-Rahman Mansour al-Khāzini or simply Abu
al-Fath Khāzini (Arabic: أبو الفتح الخازني, Persian: ابولفتح خازنی)
(flourished 1115–1130) was a Muslim
astronomerof Persian Greek ethnicity
from Merv,
then in the Khorasan province
of Persia (located
in today's Turkmenistan). Merv was known for its literary
and scientific achievements.[1]
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Title
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Al-Khazini
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Born
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11th century
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Died
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12th century
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Ethnicity
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Era
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Creed
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Main interest(s)
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Influenced by[show]
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Life
Al-Khazini
was a slave in Marw.[2] He
was the pupil of Umar Khayyam.[2] He
got his name from his master al-Khanzin. His master is responsible for his
education in mathematics and philosophy.[1][2] Al-Khazini
was known for being a humble man. He refused thousands of Dinar for his works, saying he did not need
much to live on because it was only his cat and himself in his household.[1]Al-Khazini is one of the few Islamic
astronomers to be known for doing original observations.[1] His
works are used and very well known in the Islamic world, but very few other
places around the world acknowledge his work.[1]
Achievements
Al
Khazini seems to have been a high government official under Sanjar ibn
Malikshah and the sultan of the Seljuk Empire. He did most of his work in
Merv, where they are known for their libraries.[1] His
best-known works are "The Book of the Balance of Wisdom",
"Treatise on Astronomical Wisdom", and "The Astronomical Tables
for Sanjar".[1]
"The
Book of the Balance of Wisdom" is an encyclopedia of medieval mechanics
and hydrostatics composed of eight books with fifty chapters.[1] It
is a study of the hydrostatic balance and the ideas behind statics and
hydrostatics, it also covers other unrelated topics.[1] There
are four different manuscripts of "The Book of the Balance of Wisdom"
that have survived.[1] The
balance al-Khazini built for Sanjar’s treasury was modeled after the balance
al-Asfizari, who was a generation older than al-Khazini, built.[1] Sanjar’s
treasurer out of fear destroyed al-Asfizari’s balance; he was filled with grief
when he heard the news.[1] Al-Khazini
called his balance "combined balance" to show honor towards
Al-Asfizari.[1] The
meaning of the balance was a "balance of true judgment".[1] The
job of this balance was to help the treasury see what metals were precious and
which gems were real or fake.[1] In
"The Book of the Balance of Wisdom" al-Khazini states many different
examples from the Koran ways that his balance fits into religion.[1] When
al-Khazini explains the advantages of his balance he says that it
"performs the functions of skilled craftsmen", its benefits are
theoretical and practical precision.[1]
The
"Treatise on Astronomical Wisdom" is a relatively short work.[1] It
has seven parts and each part is assigned to a different scientific instrument.[1] The
seven instruments include: a triquetrum, a dioptra, a "triangular
instrument," a quadrant, devices involving reflection, an astrolabe, and
simple tips for viewing things with the naked eye.[1] The
treatise describes each instrument and their uses.[1]
"The
Astronomical Tables for Sanjar" is said to have been composed for Sultan
Sanjar, the ruler of Merv and his balance was made for Sanjar’s treasury.[1] The
tables in "The Astronomical Tables for Sanjar" are tables of
holidays, fasts, etc.[1] The
tables are said to have the latitudes and longitudes of forty-three different
stars, along with their magnitudes and (astrological) temperaments.[1] It
is said that al-Khazini’s observations for this work were probably done in Merv
in various observatories with high quality instruments.[1]
Abu Jafar al-Khazin may have worked on both astronomy
and number theory or
there may have been two mathematicians both working around the same period, one
working on astronomy and one on number theory. As far as this article is
concerned we will assume that al-Khazin worked on both topics. There seems no
way of being certain which position is correct.
Al-Khazin's
family were from Saba, a kingdom in southwestern Arabia, perhaps better known
as Sheba from the biblical story of King Solomon and the Queen of Sheba. In the Fihrist, a tenth century survey of Islamic
culture, he is described Al-Khurasani which means that he came from Khurasan in
eastern Iran.
The
Buyid dynasty, ruling in western Iran and Iraq, reach its peak around the time
that al-Khazin lived. It undertook public schemes such as building hospitals
and dams, as well as patronising the arts and sciences. Rayy, situated
southeast of present day Tehran, was one of the major cultural centres of the
Buyid dynasty. Islamic writers described Rayy as:-
... a city of extraordinary beauty, built largely of
fired brick and brilliantly ornamented with blue faience (glazed
earthenware).
Al-Khazin
was one of the scientists brought to the court in Rayy by the ruler of the
Buyid dynasty, Adud ad-Dawlah, who ruled from 949 to 983. We know that in
959/960 al-Khazin was required by the vizier of Rayy, who was appointed by Adud
ad-Dawlah, to measure the obliquity of the ecliptic (the angle which the plane in which
the sun appears to move makes with the equator of the earth). He is said to
have made the measurement:-
... using a ring of about 4 meters.
One
of al-Khazin's works Zij
al-Safa'ih (Tables of the
disks of the astrolabe) was described by his successors as the best work
in the field and they make many reference to it. The work describes some
astronomical instruments, in particular it describes an astrolabe fitted with
plates inscribed with tables and a commentary on the use of these. A copy of
this instrument was made but vanished in Germany at the time of World War II. A
photograph of this copy was taken and the article [5]
examines this.
Al-Khazin
wrote a commentary on Ptolemy's Almagest which was criticised by al-Biruni for being too verbose. Only one
fragment of this commentary has survived and a translation of it is given in [6].
The fragment which has survived contains a discussion by al-Khazin of Ptolemy's
argument that the universe is spherical. Ptolemy wrote [6]:-
.. of different figures of equal perimeter, the one
with more angles is greater in capacity, and therefore it is necessary that a
circle is the greatest of surfaces (i.e. of all plane
figures with a constant perimeter) and
the sphere the greatest of solids.
Al-Khazin
gives 19 propositions relating to this statement by Ptolemy.
The most interesting results show, with a very ingenious proof, that an
equilateral triangle has a greater area than any isosceles or scalene triangle
with the same perimeter. When he tries to generalise this result to polygons,
however, al-Khazin gives incorrect proofs. Other results among the 19 are based
on propositions given byArchimedes in On
the sphere and cylinder. The author of [6]
argues that the ingenious results on triangles are unlikely to be due to
al-Khazin but are probably taken by him from some unknown source.
The
suggestion in [6]
that al-Khazin is a third rate mathematician is somewhat doubtful given his
work on number theory but as we stated at the beginning of this article, it is
possible that there were two mathematicians of the same name. The papers [4],
[9]
and [7]
all look at this number theory work by al-Khazin (see also [2]
and [3]).
The work of al-Khazin which is described seems to have been motivated by work
of a mathematician by the name of al-Khujandi.
Al-Khujandi claimed to have proved that x3 + y3 = z3 is impossible for whole numbers x, y, z which of course is the n = 3 case of Fermat's Last Theorem. In a letter al-Khazin
wrote:-
I demonstrate earlier ... that what Abu Muhammad al-Khujandi advanced - may God have mercy on him -
in his demonstration that the sum of two cubic numbers is not a cube is
defective and incorrect.
This
seems to have motivated further correspondence on number theory between
al-Khazin and other Arabic mathematicians. Results by al-Khazin here are
interesting indeed. His main result is to:-
... show how, if we are given a number, to find a square number so
that if the given number were added to it or subtracted from it the result
would be square.
In
modern notation the problem is given a natural number a, find natural numbers x, y, z so that x2 + a = y2 and x2 - a = z2.
Al-Khazin proves that the existence of x, y, z with these properties is equivalent to
the existence of natural numbers u, v with a = 2uv, and u2 + v2 is a square (in fact u2 + v2 = x2).
The smallest example of a satisfying these conditions is 24 which al-Khazin
gives
52 +
24 = 72, 52 -
24 = 12.
He
also gives a = 96 with
102 +
96 = 142, 102 -
96 = 22
although,
rather strangely, he seems to discount this case by another of his statements.
Rashed suggests this may be because 96 = 2 × 48 = 2 × 6 × 8 and 62 + 82 = 102is not a primitive
Pythagorean triple.
There
is a mystery which Rashed notes in [7]
(also in [2]
and [3]).
This relates to the quote above by al-Khazin regarding the false proof by al-Khujandi of the impossibility of proving x3 + y3 = z3. Rashed has discovered
a manuscript which appears to be by al-Khazin, yet contains exactly what he had
attributed to al-Khujandi.
Although al-Khazin could have realised the error in al-Khujandi's
proof and attempted a similar proof himself which he believed correct, there is
no really satisfactory explanation of these facts.
Finally
we should mention that al-Khazin proposed a different solar model from that of Ptolemy. Ptolemy had the sun moving in uniform circular
motion about a centre which was not the earth. Al-Khazin was unhappy with this
model since he claimed that if this were the case then the apparent diameter of
the sun would vary throughout the year and observation showed that this were
not the case. Of course the apparent diameter of the sun does vary but by too
small an amount to be observed by al-Khazin. To get round this problem,
al-Khazin proposed a model in which the sun moved in a circle which was centred
on the earth, but its motion was not uniform about the centre, rather it was
uniform about another point (called the excentre).
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