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The Binomial Theorem Described by Amir Kalan Al-Bukhari, Circa 1297 A.D.

In a text by al-Karaji (beg. of 11th c.), reported by al-Samaw'al (d. I 175) in his al-Bdhir,"1" one finds the table of binomial coefficients, its formation law Cmn = Cm-1n-1+ Cmn=1, and the expansion (a + b)n =
for integer n. R. Rashed believes this to be the first known m=o text in which the binomial theorem is elaborated. "2"

In an Arabic manuscript on mathematics, written approximately three hundred years later, I came across a page which explains the expansion of the binomial theorem clearly and vividly up to the seventh power for the sum of two quantities. Furthermore, the author urges the reader to follow the same procedure for all the other powers.

The text is entitled Balance of Equation in the Science of Algebra and Comparison (Qustas al-Mu`adalah fi Ilm al jabr wa al-Muqabalah) and dated 696 A.H.'I297 A.D. In his introduction to his text, the author, Amir Kalan Ibn Amir Mohammad Ibn Amir Imam al-Bukhari, describes himself saying he had studied and practiced mathematics for many years and had dictated several treatises containing original ideas. He complains that his ideas were made well known to students, some of whom plagiarized his work, claiming it as their own.

Amir Kalan is unknown to present day scholars. I have been unable to definitely identify him since he does not write his actual name but his title. The most likely identification is that Amir Kalan is Mohammad Ibn Mubarakshah Shams al-Din Mirak al-Bukhari al-Harawi. "3" My conclusions concerning the identity of Amir Kalan are based on the following:
1. Mirak al-Bukhari died c. 1339 A.D. The date of the writing of the manuscript indicates that it was written some 42 years before Mirak's death.
2. The name Amir Kalan al-Bukhari indicates that the author of the manuscript was a Persian. Yet the manuscript is written in Arabic. Sarton maintains that Mirak al-Bukhari was a Persian philosopher and astronomer writing in Arabic. He further asserts that little is known about him except that he came from or flourished in Bukhara or Harat.
3. Mirak means `little lord', a title given to him, quite possibly, by his father, the Amir. It is quite believable that such a title would be changed with maturity to Amir Kalan, which means `great lord.'
One can not conclude positively that Mirak al-Bukhari, the `little lord' of Bukhara, is indeed Amir Kalan al-Bukhari, the `great lord' of Bukhara. Perhaps in the future, one may be able to speak more confidently of the identity of the man.
The manuscript was found in Kara Mustafa Library in Istanbul, along with Abu Kamil's Kitab fi al Jabr wa al-Muqabalah,"4" by the late Dr. Martin Levey. This manuscript is in large hand writing with, mostly, twenty-one lines on a page. It is divided into ten chapters dealing with addition, subtraction, multiplication, division, fractions, powers, etc. The total number of pages in the entire manuscript is 230 pages.
The text has no mathematical notations and is completely verbal and rhetorical. There are no numerals, no vowels, no commas, or any other punctuations. In the fashion characteristic of his predecessors, alKhawarazmi and Abu Kamil, he calls an unknown a `thing'. In the text, the square of the thing is called mal. Cube is referred to as ka`b. X4 is called mal mal. In the same way mal ka`b is X5, ka'b ka'b is X6, and mal mal ka'b is X7. For convenience, I shall use the modern terminology.
The book exhibits a systematic approach to the subjects treated, i.e. he uses definitions, theorems, and generalizations. Yet, there are no formal proofs of any of the problems treated. Amir Kalan al-Bukhari acknowledges this in the beginning of the book, requesting students not to be concerned with this point. He maintains that their place is in the science of geometry. He prays that God Almighty would make it possible for him to write another book in which he would furnish both geometrical and mathematical proofs of his problems.
Amir Kalan al-Bukhari's discussion of the binomial theorem begins at the end of chapter 6 (fols. 2 i a-26a) . At first glance, there is nothing important about this chapter. It looks as though Amir Kalan al-Bukhari is trying to explain how to find square roots of numbers and algebraic expressions such as 16 or X4. However, upon closer examination one can clearly see that the author is aware of the expansion of the binomial theorem.
Amir Kalan, who in other parts of his text gives us the expansion of
(a + b)2, begins on line four of fol. 25b, at the end of the chapter, to give us a detailed exposition of the binomial theorem. He writes, `We shall end this chapter by discussing the procedure for the expansion of algebraic expressions. This will enable us to factor such expressions in case we need to. Thus, the cube "5"of the sum is equal to the cubes of each of the two terms plus the product of each one by the square of the other taken thrice."6" The fourth power of the sum is equal to the fourth power of each term plus the product of each by the cube of the other taken four times plus the product of the square of one by the square of the other taken six times"7". The fifth power of the sum is equal to the fifth power of each term plus the product of the fourth power of each one by the other taken five times plus the product of the third power of each term by the square of the other taken ten times."8" The sixth power of the sum is equal to the sixth power of each of them and the product of each term by the fifth power of the other taken six times and the product of the fourth power of each term by the square of the other taken five"9" times and the product of the cube of each one by the cube of the other taken twenty times"10". The seventh power of the sum is equal to the seventh power of each of the terms plus the product of each one by the sixth power of the other taken seven times plus the product of the square of each term by the fifth power of the other taken twenty-one times plus the product of the cube of each term by the fourth power of the other taken thirty-five times."11"
Concluding the chapter, he writes, `We have concerned ourselves with the expression consisting of two terms because those which consist of three, four, or more terms are nothing more than special cases of two terms. Do you not see that if you want to find the cube of a three term expression you combine two of them into one. That is, combine the first two and raise it to the third power. Also raise the third term itself to a cube. Multiply the third term by the square of the sum of the first two thrice. Then multiply the sum of the first two by the square of the third thrice. The sum is the final answer to the original one. Follow the same procedure for all the other powers.'
Al-Bukhari's treatment of this subject is similar in style and content to that of al-Karaji. As an example, I will quote al-Karaji's explanation of the expansion of (a + b)5 : `son quadrato-cube est egal a la somme des quadrato-cubes de chacune de ses parties, cinq fois le produit de chacune des parties par le carre-carre de 1'autre et dix fois le produit due carre de chacune d'alles par le cube de 1'autre."12"
Al-Samaw'al gives us al-Karaji's description of what is commonly known as Pascal's triangle. Furthermore, he uses a `slightly old-fashioned form of mathematical induction "13" to demonstrate the expansion of (a + b)n and other propositions. Amir Kalan al-Bukhari makes no reference to Pascal's triangle at all, although his treatment of the binomial theorem suggests that he was well aware of it.
It is my opinion that the casual reference by Amir Kalan to this subject at the end of chapter 6 and his remark that the reader should `follow the same procedure for all the other powers' make it clear that the knowledge of the binomial theorem and the binomial coefficients was a prevalent one.

R. Rashed and A. Ahmad, Valg8re al-Bahir d'al-Samaw'al (Damascus, 1972).
2) R. Rashed, °L'induction rnath~madque: al- Karaji, as-Samaw'al', Archive for History of Exact Sciences, i, 1972, I-2I.
3) George Sarton, Introduction to The History of Science, VIII, 1947, p. 699.
4) Martin Levey, The Algebra of Abu Kamil, Winsconsin, 1966.
5) The word `cube' is missing
6) (a+b)3 = a3 + 3a2b + 3ab2 + b3.
7) (a+b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
8) (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5.
9) This is an obvious arithmetical error, for coefficient should be 15 not 5
10) (a + b)6 = a6 + 6a5b + 15a4b2 + 20a3b3 + 15a2b4 + 6ab6 + b6.
11) (a+b)7 = a7 + 7a6b + 21a5b2 + 35a4b3 + 35a3b4 + 21a2b5 + 7ab6 + b7
12) Rashed, op. cit., p.5.
13) R. Rashed, Al-Karaji, Dictionary of Scientific Biography, VII, pp. 24off.

The Islamic Quarterly, London
January - June 1978


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