This does not sound like the contents of an algebra text and indeed only the first part of the book is a discussion of what we would today recognise as algebra.
However it is important to realise that the book was intended to be highly practical and that algebra was introduced to solve real life problems that were part of everyday life in the Islam empire at that time.
Early in the book al-Khwarizmi describes the natural numbers in terms that are almost funny to us who are so familiar with the system, but it is important to understand the new depth of abstraction and understanding here
When I consider what people generally want in calculating, I found that it always is a number. I also observed that every number is composed of units, and that any number may be divided into units. Moreover, I found that every number which may be expressed from one to ten, surpasses the preceding by one unit: afterwards the ten is doubled or tripled just as before the units were: thus arise twenty, thirty, etc. until a hundred: then the hundred is doubled and tripled in the same manner as the units and the tens, up to a thousand; ... so forth to the utmost limit of numeration.
Having introduced the natural numbers, al-Khwarizmi introduces the main topic of this first section of his book, namely the solution of equations.
His equations are linear or quadratic and are composed of units, roots and squares. For example, to al-Khwarizmi a unit was a number, a root was x, and a square was x2. However, although we shall use the now familiar algebraic notation in this article to help the reader understand the notions, Al-Khwarizmi's mathematics is done entirely in words with no symbols being used.
He first reduces an equation (linear or quadratic) to one of six standard forms:
1. Squares equal to roots.
2. Squares equal to numbers.
3. Roots equal to numbers.
4. Squares and roots equal to numbers;
e.g. x2 + 10 x = 39.
5. Squares and numbers equal to roots;
e.g. x2 + 21 = 10 x.
6. Roots and numbers equal to squares;
e.g. 3 x + 4 = x2.
The reduction is carried out using the two operations of al-jabr and al-muqabala. Here "al-jabr" means "completion" and is the process of removing negative terms from an equation.
For example, using one of al-Khwarizmi's own examples, "al-jabr" transforms x2 = 40 x - 4 x2 into 5 x2 = 40 x. The term "al-muqabala" means "balancing" and is the process of reducing positive terms of the same power when they occur on both sides of an equation.
For example, two applications of "al-muqabala" reduces 50 + 3 x + x2 = 29 + 10 x to 21 + x2 = 7 x (one application to deal with the numbers and a second to deal with the roots). Al-Khwarizmi then shows how to solve the six standard types of equations.
He uses both algebraic methods of solution and geometric methods.
For example to solve the equation x2 + 10 x = 39 he writes :-
... a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned.
Now the roots in the problem before us are 10.
Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64.
Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3.
The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square.
The geometric proof by completing the square follows. Al-Khwarizmi starts with a square of side x, which therefore represents x2 (Figure 1).
To the square we must add 10x and this is done by adding four rectangles each of breadth 10/4
and length x to the square (Figure 2).
Figure 2 has area x2 + 10 x which is equal to 39.
We now complete the square by adding the four little squares each of area 5/2 x 5/2 = 25/4.
Hence the outside square in Fig 3 has area 4 x 25/4 + 39 = 25 + 39 = 64.
The side of the square is therefore 8.
But the side is of length 5/2 + x + 5/2 so x + 5 = 8, giving x = 3.
Al-Khwarizmi also wrote a treatise on Hindu-Arabic numerals. The Arabic text is lost but a Latin translation, Algoritmi de numero Indorum in English Al-Khwarizmi on the Hindu Art of Reckoning gave rise to the word algorithm deriving from his name in the title.
Unfortunately the Latin translation (translated into English in is known to be much changed from al-Khwarizmi's original text (of which even the title is unknown).
The work describes the Hindu place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0.
The first use of zero as a place holder in positional base notation was probably due to al-Khwarizmi in this work.
Methods for arithmetical calculation are given, and a method to find square roots is known to have been in the Arabic original although it is missing from the Latin version. Toomer writes :-
... the decimal place-value system was a fairly recent arrival from India and ... al-Khwarizmi's work was the first to expound it systematically.
Thus, although elementary, it was of seminal importance.
Seven twelfth century Latin treatises based on this lost Arabic treatise by al-Khwarizmi on arithmetic are discussed in
Another important work by al-Khwarizmi was his work Sindhind zij on astronomy. The work, described in detail in , is based in Indian astronomical works :-
... as opposed to most later Islamic astronomical handbooks, which utilised the Greek planetary models laid out in Ptolemy's "Almagest"...
The Indian text on which al-Khwarizmi based his treatise was one which had been given to the court in Baghdad around 770 as a gift from an Indian political mission.
There are two versions of al-Khwarizmi's work which he wrote in Arabic but both are lost.
There is also a Latin version of the longer version and both these Latin works have survived. The main topics covered by al-Khwarizmi in the Sindhind zij are calendars; calculating true positions of the sun, moon and planets, tables of sines and tangents; spherical astronomy; astrological tables; parallax and eclipse calculations; and visibility of the moon.
Al-Khwarizmi wrote a major work on geography which give latitudes and longitudes for 2402 localities as a basis for a world map.
A number of minor works were written by al-Khwarizmi on topics such as the astrolabe, on which he wrote two works, on the sundial, and on the Jewish calendar.
He also wrote a political history containing horoscopes of prominent persons.
We have already discussed the varying views of the importance of al-Khwarizmi's algebra which was his most important contribution to mathematics. Let us end this article with these words:-
In the foremost rank of mathematicians of all time stands Al-Khwarizmi. He composed the oldest works on arithmetic and algebra. They were the principal source of mathematical knowledge for centuries to come in the East and the West. The work on arithmetic first introduced the Hindu numbers to Europe, as the very name algorism signifies; and the work on algebra ... gave the name to this important branch of mathematics in the European world
