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
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
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
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
(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.
1) R. Rashed and A. Ahmad, Valg8re al-Bahir d'al-Samaw'al
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,
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