History of Astaneh
In 191 Hejri Shamsi (813 AD), during the reign of Ma’mun the Abbasid caliph, Aboo Dolaph from the Arab tribe of "Ajali" became the <occupational> governor of Astaneh. He remained the governor of Astaneh during the reign of Ma’mun’s brother, Mu'tasim too.
At that time, Astaneh was the capital of the "Jebal" (koohestan, in Arabic) territory, which its borders extended from Hamedan to Esfahan, khoozestan, and Ray. This area has been referred to as "Mad-e bozorg", The great Mede territory.
Later Ali Emadol doleh Deilami, the oldest of the 3 Deilami brothers, of the Booye Dynasty was appointed the governor of Astaneh by the Ziyarid Mardavidj in early 300 Hejri shamsi (920 AD). This is Perhaps the beginning of an end to the Arab rule of this part of Iran.
From here, Ali took hold of Esfahan after defeating the governor of Isfahan’s 4,000 men, whereas Ali only had at his disposal merely 700 men. Angry, Mardavij sent against Isfahan his son Voshmgir. Ali withdrew towards the west, but again had the opportunity to carry off a great victory against the governor of the province of Fars who was directly picked by the Abbasid caliph.
Nearly 600 years later, In 908 AH (over 500 years ago), Shah Ismail Saffavid ordered to carefully repair the mausoleum (Astaneh) of Imamzadeh Sahl-e-bn Ali in “Karaj”. Since then Karaj has been called Astaneh.
Historically, the region goes a long way back. The famous town of Hegmataneh was built not far from here. There is enough evidence to suggest that Astaneh was a military base for the Median troops.
The ancient Zoroastrian place of pilgrimage known as Shah Zand is located 4 miles to the North of Astaneh.
Astaneh has been destroyed three times, once by the mongoles, another time by an earthquake, and the last time by the Afghan invadores on their way to Isfahan.
For more information about the history of Astaneh and the region refer to the book of Karajnameh, by late Ebrahim Dehgan.
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The following is parts of a letter that one of the elite of Astaneh, Haji Molla Mohammad Bagher Abbasi, also known as "haji Akhoond" wrote to Reza Shah at the beginning of his reign, and requested help in restoring the "Emamzadeh".
There is a short history of Astaneh in his well-written letter. It shows that he was a very intelligent man, and had great knowledge of history of his hometown. God may bless his soul.
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This picture shows the last page of a book, written in 1217 Hejri Ghamari (1801) by Haji Ali Akbar Zooleh, son of Haji Pir Ahmad Japelaghi, the ancestor of the Ahmadi family in astaneh.
He died in 1274 Ghamari, (1857 AD), 56 years after this book was written in Dareh Zooleh (see his headstone below)
Amongst the other great people from this Family are: Mirazi Ali Mohamad Ahmadi (died and burried in Astaneh in 1342 (1961)), Karbalaei Ali Akbar Ahmadi (died and burried in Astaneh in 1342 (1961) and Mirza Taher Ahmadi (died within a few years of the other two, he was rather young). |
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This is the head stone of the late Haji Ali Akbar Zooleh who died in 1274 HG (1857). He was the son of the late Haj Pir Ahmad Japelaghi, also known as "the Khajeh Ahmade Bozorg" (Khajeh Ahmad the great).
THe ruins of the house they lived in still exists in the village of Zooleh 25 miles South od Astaneh. Also there is an orchard which it is still called the "bagh-e Hajji".
Picture taken by A. Shafie, in summer of 1998 in village of Zooleh |
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A mathematician from Astaneh
Al-Karaji, Abu Bekr ibn Muhammad ibn al-Husayn (953 - ca. 1029).
Al-Karaji's work centered around algebra and polynomials, giving rules for arithmetic operations to manipulate polynomials. Woepcke describes his work as introducing the "theory of algebraic calculus". Stemming from this, Al-Karaji investigated binomial coefficients and Pascal's triangle. Additionally, Al-Karaji used induction to prove his results.
We know that al-Karaji lived in Baghdad for most of his life and that his chief mathematical works were written during the time when he lived in that city. His important treatise on algebra Al-Fakhri was dedicated to the ruler of Baghdad and was written in the city. However, at some later point in his career, al-Karaji left Baghdad to live in what are described as the "mountain countries". He seems to have given up mathematics at this time and concentrated on engineering topics such as the drilling of wells.
The importance of al-Karaji in the development of mathematics is viewed rather differently by different authors. The reason for this, rather in the same spirit as the different views on al-Khwarizmi, depends on the significance one attaches to the style of his mathematics. Some consider that his work is merely reworking ideas from earlier mathematicians while others see him as the first person to completely free algebra from geometrical operations and replace them with the arithmetical type of operations which are at the core of algebra today.
Crossley [3] sounds relatively unimpressed by al-Karaji's contributions (although he describes the content accurately):-
[Al-Karaji] gives rules for the arithmetic operations including (essentially) the multiplication of polynomials. ... al-Karaji usually gives a numerical example for his rules but does not give any sort of proof beyond giving geometrical pictures. Often he explicitely says that he is giving a solution in the style of Diophantus. He does not treat equations above the second degree except for ones which can easily be reduced to at most second degree equations followed by the extraction of roots. The solutions of quadratics are based explicitly on the Euclidean theorems ...
Woepcke was the first historian to realise the importance of al-Karaji's work and later historians mostly agree with his interpretation. He describes it as the first appearance of a:-
... theory of algebraic calculus ... .
Rashed (see [5] which contains Rashed's article from [1] and other writings by Rashed on al-Karaji) agrees with Woepcke's interpretation and perhaps goes even further in stressing al-Karaji's importance. He writes:-
... the more-or-less explicit aim of
To give another quote from Rashed's description of al-Karaji's contribution:-
Al-Karaji's work holds an especially important place in the history of mathematics. ... the discovery and reading of the arithmetical work of
So what was this new departure in algebra? Perhaps it is best described by al-Samawal, one of al-Karaji's successors, who described it as [5]:-
... operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.
What al-Karaji achieved in Al-Fakhri was first to define the monomials x, x2, x3, ... and 1/x, 1/x2, 1/x3, ... and to give rules for products of any two of these. So what he achieved here was defining the product of these terms without any reference to geometry. In fact he almost gave the formula
xnxm = xmfor all integers n and m +n
but he failed to make the definition x0 = 1 so he fell just a little short.
Having given rules for multiplication and division of monomials al-Karaji then looked at "composite quantities" or sums of monomials. For these he gave rules for addition, subtraction and multiplication but not for division in the general case, only giving rules for the division of a composite quantity by a monomial. He was able to give a rule for finding the square root of a composite quantity which is not completely general since it required the coefficients to be positive, but it is still a remarkable achievement.
Al-Karaji also uses a form of mathematical induction in his arguments, although he certainly does not give a rigorous exposition of the principle. Basically what al-Karaji does is to demonstrate an argument for n = 1, then prove the case n = 2 based on his result for n = 1, then prove the case n = 3 based on his result for n = 2, and carry on to around n = 5 before remarking that one can continue the process indefinitely. Although this is not induction proper, it is a major step towards understanding inductive proofs.
One of the results on which al-Karaji uses this form of induction comes from his work on the
Let us now recall a principle for knowing the necessary number of multiplications of these degrees by each other, for any number divided into two parts. Al-Karaji said that in order to succeed we must place 'one' on a table and 'one' below the first 'one', move the first 'one' into a second column, add the first 'one' to the 'one' below it. Thus we obtain 'two', which we put below the transferred 'one' and we place the second 'one' below the 'two'. We have therefore 'one', 'two', and 'one'.
To see how the second column of 1,2,1 corresponds to squaring a+b al-Samawal continues to describe Al-Karaji's work writing:-
This shows that for every number composed of two numbers, if we multiple each of them by itself once - since the two extremes are 'one' and 'one' - and if we multiply each one by the other twice - since the intermediate term is 'two' - we obtain the square of this number.
This is a beautiful description of the binomial theorem using the Pascal triangle. The description continues up to the binomial coefficients which give (a+b)5 but we shall only quote how al-Karaji constructs the third column from the second:-
If we transfer the 'one' in the second column into a third column, then add 'one' from the second column to 'two' below it, we obtain 'three' to be written under the 'one' in the third column. If we then add 'two' from the second column to ''one' below it we have 'three' which is written under the 'three', then we write 'one' under this 'three'; we thus obtain a third column whose numbers are 'one', 'three', 'three', and 'one'.
The table al-Karaji constructed looks like the Pascal triangle on its side.
Other results obtained by al-Karaji include summing the first n natural numbers, the squares of the first n natural numbers and the cubes of these numbers. He proved that the sum of the first n natural numbers was n(1/2 + n/2). He also gave (in Rashed and Ahmad's translation, see for example [5]):-
The sum of the squares of the numbers that follow one another in natural order from one is equal to the sum of these numbers and the product of each of them by its predecessor.
Al-Karaji also considered sums of the cubes of the first n natural numbers writing (in Rashed and Ahmad's translation, see for example [5]):-
If we want to add the cubes of the numbers that follow one another in their natural order we multiply their sum by itself.
Al-Karaji showed that (1 + 2 + 3 + ... + 10)2was equal to 13 + 23 + 33 + ... + 103. He did this by first showing that (1 + 2 + 3 + ... + 10)2 = (1 + 2 + 3 + ... + 9)2 + 103. He could now use the same rule on (1 + 2 + 3 + ... + 9)2, then on (1 + 2 + 3 + ... + 8)2 etc. to get
( 1 + 2 + ... + 10)2
= (1 + 2 + 3 + ... + 8)2 + 93 + 103
= (1 + 2 + 3 + ... + 7)2 + 83 + 93 + 103
= . . .
= 13 + 23 + 33 + ... + 103.
Finally we should mention the influence of Diophantus on al-Karaji. The first five books of Diophantus's Arithmetica had been translated into Arabic by ibn Liqa around 870 and these were studied by al-Karaji. Woepcke in his introduction to Al-Fakhri ([7] or [8]) writes that he found:-
... more than a third of the problems of the first book of
Al-Karaji also invented many new problem of his own but even those of Diophantus were certainly not just taken without further development. He always tried to generalise Diophantus's results and to find methods which were more generally applicable.
It was not only to algebra that al-Karaji contributed. The paper [9] discusses some of his geometrical work. This occurs in a chapter entitled On measurement and balances for measuring of buildings and structures. al-Karaji defines points, lines, surfaces, solids and angles. He also gives rules for measuring both plane and solid figures, often using arches as examples. He also gives methods of weighing different substances.
Article by:J J O'Connor and E F RobertsonDiophantus, the problems of the second book starting with the eighth, and virtually all the problems of the third book were included by al-Karaji in his collection. binomial theorem, the binomial coefficients and the Pascal triangle. In Al-Fakhri al-Karaji computed (a+b)3 and in Al-Badi he computed (a-b)3 and (a+b)4. The general construction of the Pascal triangle was given by al-Karaji in work described in the later writings of al-Samawal. In the translation by Rashed and Ahmad (see for example [5]) al-Samawal writes:- Diophantus, in the light of the algebraic conceptions and methods of al-Khwarizmi and other Arab algebraists, made possible a new departure in algebra by Al-Karaji ... [al-Karaji's] exposition was to find the means of realising the autonomy and specificity of algebra, so as to be in a position to reject, in particular, the geometric representation of algebraic operations.
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Another article about Karaji
[al-] Karaji, Abu Bekr ibn Muhammad ibn al-Husayn (early 10th century AD). The advanced knowledge of the Persians concerning groundwater is demonstrated by a recently discovered book by Mohammed Karaji titled: The Extraction of Hidden Waters. This book reveals a profound and exacting technical understanding of groundwater theory and is the oldest known text on the subject. Karaji's knowledge of groundwater, which surpassed that of western scholars 7 centuries later, is in general agreement with modern understanding of the subject. For example, Karaji was familiar with the general concepts of the hydrological cycle. While he never featured the whole cycle as we know it, he records in different passages of his book each individual phase.
Karaji spent most of his life in Bagdad, working mainly as a mathematician. He wrote several books on algebra and geometry which were translated into German during the 19th century. These translations introduced Karaji to western scholars where he is now counted with the classical Arabic mathematicians. At an advanced age, Karaji returned to the central highlands of his native Persia. As a means of earning his subsistence, he published Extraction.
Karaji and other Persian scholars were familiar with the basic hydrologic, geologic, and engineering principles associated with groundwater. Karaji himself exhibited extensive skills and expertise regarding: (1) the classification of soils, (2) the search for fresh water, and (3) the different types and hydraulic characteristics of aquifers. He pioneered work on the use of plant growth as an indicator of groundwater aquifers, and invented ingenious devices employed in surveying and tunneling. Much of the Karaji's book deals with the techniques of exploring for groundwater, mainly how to dig wells and qanats. The methods he describes are still used in many parts of the Middle East and Asia.
References: Pazwash, Hormoz and Gus Mavrigian, 1980, "A Historical Jewelpiece-Discovery of the Millennium Hydrological Works of Karaji," Water Resources Bulletin, December, pp. 1094-1096; Nadji, Mehdi and Rudolf Voight, 1972, "'Exploration for Hidden Water' by M. Karaji-Oldest Textbook on Hydrology?" Groundwater, September-October, pp. 43-46.
Emamzadeh Sahl-ebn-e Ali, picture taken in Summer of 2003 (1382) By Hossein Abolhasani
