1256 - 1321

Mathematics, Medicine

**
Ibn Al-Banna** is also
known as Abu'l-Abbas Ahmad ibn Muhammad ibn Uthman al-Azdi. It
is a little unclear whether al-Banna was born in the city of
Marrakesh or whether it was the region of Marrakesh which was
named Morocco by Europeans. There is a claim that al-Banna was
born in Granada in Spain and moved to North Africa for his
education. What is certain is that he spent most of his life in
Morocco.

The Marinids
tribe were allies of the Umayyad caliphs of Córdoba. The tribe
lived in eastern Morocco then, under their ruler Abu Yahya, they
began to conquer the region. The Marinids captured Fez in 1248
and made it their capital. They captured Marrakesh from the
ruling Almohads tribe in 1269, thus taking control of the whole
of Morocco. Having conquered Morocco, the Marinids tried to help
Granada to prevent the Christian advance through their country.
The strong link between Granada and Morocco may account for the
confusion as to which country al-Banna was a native.

Morocco was
certainly the country that al-Banna was educated in, learning
the leading mathematical skills of the period. He studied
geometry in general, and Euclid's *Elements* in particular.
He also studied fractional numbers and learnt much of the
impressive contributions that the Arabs had made to mathematics
over the preceding 400 years. The Marinids had a strong culture
for learning and Fez became their centre of learning. At the
university in Fez Al-Banna taught all branches of mathematics,
which at this time included arithmetic, algebra, geometry and
astronomy. Fez was a thriving city with a new quarter being
built housing the Royal Palace and the adjoining Great Mosque.
Many students studied under al-Banna in this thriving academic
community.

It is clear
that al-Banna wrote a large number of works, in fact 82 are
listed by Renaud (see for example [9]). Not all are on
mathematics, but the mathematical texts included an introduction
to Euclid's *Elements*, an algebra text and various works
on astronomy. One difficulty with the works on mathematics is
knowing how much of the material which al-Banna presents is
original and how much is simply his version of work by earlier
Arab mathematicians which has been lost. We should certainly say
that al-Banna does not claim any originality and, indeed, the
style of his writing would suggest that he is collecting
together ideas that he has learnt from other mathematicians.

Two "firsts"
for al-Banna are that he seems to have been the first to
consider a fraction as a ratio between two numbers (see [12] for
more details) and he is the first to use the expression almanakc
(in Arabic al-manakh meaning weather) in a work containing
astronomical and meteorological data.

Perhaps al-Banna's
most famous work is *Talkhis amal al-hisab* (Summary of
arithmetical operations) and the *Raf al-Hijab* which is
al-Banna's own commentary on the *Talkhis amal al-hisab*.
It is in this work that al-Banna introduces some mathematical
notation which has led certain authors to believe that algebraic
symbolism was first developed in Islam by ibn al-Banna and al-Qalasadi
(see for example [6]). We refer the reader to the biography of
al-Qalasadi where we present arguments to show that neither
al-Banna nor al-Qalasadi were the inventors of mathematical
notation.

There are,
however, many interesting mathematical ideas and results which
appear in the *Raf al-Hijab*. For example it contains
continued fractions and they are used to compute approximate
square roots. Other interesting results on summing series are
the results

1^{3}
+ 3^{3} + 5^{3} + ... + (2*n*-1)^{3}
= *n*^{2}(2*n*^{2} - 1) and

1^{2}
+ 3^{2} + 5^{2} + ... + (2*n*-1)^{2}
= (2*n* + 1)2*n*(2*n* - 1)/6.

Perhaps the
most interesting of all is the work on binomial coefficients
which is described in detail in [2] and [3]. If we denote the
binomial coefficient *p* choose *k* by _{p}*C*_{k}
then al-Banna shows that

_{p}*C*_{2}
= *p*(*p*-1)/2

and then that

_{p}*C*_{3}
= _{p}*C*_{2}(*p*-2)/3.

He writes
(see for example [2] or [3]):-

*...
the ternary combination is thus obtained by multiplying the
third of the third term preceding the given number; and so
we always multiply the combination that precedes the
combination sought by the number that precedes the given
number, and whose distance to it is equal to the number of
combinations sought. From the product, we take the part that
names the number of combinations.*

Although this
is a little difficult to interpret, what al-Banna is stating
here is that

_{p}*C*_{k}
= _{p}*C*_{k-1}(*p*
- (*k* - 1) )/*k*.

He then goes
on to give the familiar (to us) result

_{p}*C*_{k}
= *p*(*p* - 1)(*p* - 2)...(*p* - *k*
+ 1)/(*k* !)

As Rashed
points out in [2], this is only a small step from the Pascal
triangle results given three hundred years earlier by al-Karaji,
then still one hundred years before al-Banna by al-Samawal.
However Rashed writes:-

*... in
our opinion, there is something more fundamental than
*[*the Pascal triangle*]* results;
it is precisely the combinatorial appearance of ibn al-Banna's
exposition, together with the relation he partially
establishes between polygonal numbers and combinations. It
concern, in the first place, triangular numbers and
combinations of p objects in twos, and then polygonal
numbers of order *4* and combinations of p objects in
threes.*

**Article by:**
*J J O'Connor* and *E F Robertson*

**November 1999**