it was various that mathematics: Februari 2008

Kamis, 28 Februari 2008

ABUL WAFA MUHAMMAD AL-BUZJANI(940-997 C.E.)

Abul Wafa Muhammad Ibn Muhammad Ibn Yahya Ibn Ismail al-Buzjani was born in Buzjan, Nishapur in 940 C.E. He flourished as a great mathematician and astronomer at Baghdad and died in 997/998 C.E. He learnt mathematics in Baghdad. In 959 C.E. he migrated to Iraq and lived there till his death.

Abul Wafa's main contribution lies in several branches of mathematics, especially geometry and trigonometry. In geometry his contribution comprises solution of geometrical problems with opening of the compass; construction of a square equivalent to other squares; regular polyhedra; construction of regular hectagon taking for its side half the side of the equilateral triangle inscribed in the same circle; constructions of parabola by points and geometrical solution of the equations:

x⁴ = a and x⁴ + ax³ = b

Abul Wafa's contribution to the development of trigonometry was extensive. He was the first to show the generality of the sine theorem relative to spherical triangles. He developed a new method of constructing sine tables, the value of sin 30' being correct to the eighth decimal place. He also developed relations for sine (a+b) and the formula:

2 sin² (a/2) = 1 - cos a , and
sin a = 2 sin (a/2) cos (a/2)

In addition, he made a special study of the tangent and calculated a table of tangents. He introduced the secant and cosecant for the first time, knew the relations between the trigonometric lines, which are now used to define them, and undertook extensive studies on conics.

Apart from being a mathematician, Abul Wafa also contributed to astronomy. In this field he discussed different movernents of the moon, and discovered 'variation'. He was also one of the last Arabic translators and commentators of Greek works.

He wrote a large number of books on mathematics and other subjects, most of which have been lost or exist in modified forms. His contribution includes Kitab 'Ilm al-Hisab, a practical book of arithmetic, al-Kitab al-Kamil (the Complete Book), Kitab al-Handsa (Applied Geometry). Apart from this, he wrote rich commentaries on Euclid, Diophantos and al-Khawarizmi, but all of these have been lost. His books now extant include Kitab 'Ilm al-Hisab, Kitab al- Handsa and Kitab al-Kamil.

His astronomical knowledge on the movements of the moon has been criticized in that, in the case of 'variation' the third inequality of the moon as he discussed was the second part of the 'evection'. But, according to Sedat, what he discovered was the same that was discovered by Tycho Brache six centuries later. Nonetheless, his contribution to trigonometry was extremely significant in that he developed the knowledge on the tangent and introduced the secant and cosecant for the first time; in fact a sizeable part of today's trigonometry can be traced back to him.

Stefan Banach: do you remember Banach Space?

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

Born: 30 March 1892 in Kraków, Austria-Hungary (now Poland)
Died: 31 Aug 1945 in Lvov, (now Ukraine)

Stefan Banach's father was Stefan Greczek. The first thing to notice is that Banach was not his father's surname, but Banach was given his father's first name. Stefan Greczek was a tax official who was not married to Banach's mother who vanished from the scene after Stefan was baptised, when he was only four days old, and nothing more is known of her. The name given as Stefan's mother on his birth certificate is Katarzyna Banach. She is thought by some to have been the servant of Stefan's mother, while others claim that she was a laundress who took care of Stefan when he was very young. In later life Banach tried to find out who his mother was but his father refused to say anything except that he had been sworn to secrecy over her identity.

Stefan Greczek was born in a small village called Ostrowsko, some 50 km south of Kraków. It was to Ostrowsko, to his grandmother's home, that Banach was taken after his baptism. However, when Banach's grandmother took ill, Stefan Greczek arranged for his son to be brought up by Franciszka Plowa who lived in Kraków with her daughter Maria. Although Banach never went back to live with his grandmother, he did visit her frequently as he grew up. Maria's guardian was a French intellectual Juliusz Mien and he quickly recognised the talents that Banach had. Mien taught the young boy to speak French and in general gave him an appreciation for education.

Banach attended primary school in Kraków, leaving the school in 1902 to begin his secondary education at the Henryk Sienkiewicz Gymnasium No 4 in Kraków. By a fortunate coincidence, one of the students in Banach's class was Witold Wilkosz who himself went on to become a professor of mathematics. The school does not appear to have been a particularly good one and in 1906 Wilkosz left to move to a better Gymnasium. Banach, however, remained at Henryk Sienkiewicz Gymnasium No 4 although he maintained contact with Wilkosz.

During his first few years at the Gymnasium Banach achieved first class grades with mathematics and natural sciences being his best subjects. A fellow school pupil recalled Banach at this period in his life (see [3]):-

[Banach] was pleasant in dealings with his colleagues, but outside of mathematics he was not interested in anything. If he spoke at all, he would speak very rapidly, as rapidly as he thought mathematically. ... Wilkosz was a similar phenomenon. Between the two of them there was no mathematical problem that they could not speedily tackle. Also, while Banach was faster in mathematical problems, Wilkosz was phenomenally fast in solving problems in physics, which were of no interest to Banach.

The excellent grades of his early years gave way to poorer grades as he approached his final school examination. He passed this examination in 1910 but he failed to achieve a pass with distinction, an honour which went to about one quarter of the students. On leaving school Banach and Wilkosz both wanted to study mathematics, but both felt that nothing new could be discovered in mathematics so each chose to work in a subject other than mathematics. Banach chose to study engineering, Wilkosz chose oriental languages. That two such outstanding future mathematicians could make a decision for this reason must mean that there was nobody to properly advise them.

Banach's father had never given his son much support, but now once he left school he quite openly told Banach that he was now on his own. Banach left Kraków and went to Lvov where he enrolled in the Faculty of Engineering at Lvov Technical University. It is almost certain that Banach, without any financial support, had to support himself by tutoring. This must have occupied quite a lot of his time and when he graduated in 1914 he had taken longer to complete the course than was normal. He had returned to Kraków frequently during the period of his studies in Lvov from 1910 to 1914. It is not entirely clear what Banach's plans were in 1914 but the outbreak of World War I in August, shortly after his graduation, saw Banach leave Lvov.

Lvov was, at the time Banach studied there, under Austrian control as it had been from the partition of Poland in 1772. In Banach's youth Poland, in some sense, did not exist and Russia controlled much of the country. Warsaw only had a Russian language university and was situated in what was named "Vistula Land". With the outbreak of World War I, the Russian troops occupied the city of Lvov. Banach was not physically fit for army service, having poor vision in his left eye. During the war he worked building roads but also spent time in Kraków where he earned money by teaching in the local schools. He also attended mathematics lectures at the Jagiellonian University in Kraków and, although this is not completely certain, it is believed that he attended Zaremba's lectures.

A chance event occurred in the spring of 1916 which was to have a major impact on Banach's life. Steinhaus, who had been undertaking military service, was about to take up a post at the Jan Kazimierz University in Lvov. However he was living in Kraków in the spring of 1916, waiting to take up the appointment. He would walk through the streets of Kraków in the evenings and, as he related in his memoirs:-

During one such walk I overheard the words "Lebesgue measure". I approached the park bench and introduced myself to the two young apprentices of mathematics. They told me they had another companion by the name of Witold Wilkosz, whom they extravagantly praised. The youngsters were Stefan Banach and Otto Nikodym. From then on we would meet on a regular basis, and ... we decided to establish a mathematical society.

Steinhaus told Banach of a problem which he was working on without success. After a few days Banach had the main idea for the required counterexample and Steinhaus and Banach wrote a joint paper, which they presented to Zaremba for publication. The war delayed publication but the paper, Banach's first, appeared in the Bulletin of the Kraków Academy in 1918. From the time that he produced these first results with Steinhaus, Banach started to produce important mathematics papers at a rapid rate. Of course it is impossible to say whether, without the chance meeting with Steinhaus, Banach would have followed the route of research in mathematics. It was also through Steinhaus that Banach met his future wife Lucja Braus. They were married in the mountain resort of Zakopane in 1920.

On Steinhaus's initiative, the Mathematical Society of Kraków was set up in 1919. Zaremba chaired the inaugural meeting and was elected as the first President of the Society. Banach lectured to the Society twice during 1919 and continued to produce top quality research papers. The Mathematical Society of Kraków went on to became the Polish Mathematical Society in 1920.

Banach was offered an assistantship to Lomnicki at Lvov Technical University in 1920. He lectured there in mathematics and submitted a dissertation for his doctorate under Lomnicki's supervision. This was, of course, not the standard route to a doctorate, for Banach had no university mathematics qualifications. However, an exception was made to allow him to submit On Operations on Abstract Sets and their Application to Integral Equations. This thesis [1]:-

... is sometimes said to mark the birth of functional analysis.

In 1922 the Jan Kazimierz University in Lvov awarded Banach his habilitation for a thesis on measure theory. The University Calendar for 1921-22 reports [3]:-

On 7 April 1922, by resolution of the Faculty Council, Dr Stefan Banach received his habilitation for a Docent of Mathematics degree. He was appointed Professor Extraordinary of that subject by decree of the Head of State issued on 22 July 1922.

In 1924 Banach was promoted to full professor and he spent the academic year 1924-25 in Paris. The years between the wars were extremely busy one for Banach. As well as continuing to produce a stream of important papers, he wrote arithmetic, geometry and algebra texts for high schools. He also was very much involved with the publication of mathematics. In 1929, together with Steinhaus, he started a new journal Studia Mathematica and Banach and Steinhaus became the first editors. The editorial policy was:-

... to focus on research in functional analysis and related topics.

Another important publishing venture, begun in 1931, was a new series of Mathematical Monographs. These were set up under the editorship of Banach and Steinhaus from Lvov and Knaster, Kuratowski, Mazurkiewicz, and Sierpinski from Warsaw. The first volume in the series Théorie des Opérations linéaires was written by Banach and appeared in 1932. It was a French version of a volume he originally published in Polish in 1931 and quickly became a classic. In 1936 Banach gave a plenary address at the International Congress of Mathematicians in Oslo. In this address he described the work of the whole of the Lvov school, and he also spoke of the plans which they had to develop their ideas further.

Another important influence on Banach was the fact that Kuratowski was appointed to the Lvov Technical University in 1927 and worked there until 1934. Banach collaborated with Kuratowski and they wrote some joint papers during this period.

The way that Banach worked was unconventional. He liked to do mathematical with his colleagues in the cafés of Lvov. Ulam recalls in [4] frequent sessions in the Scottish Café:-

It was difficult to outlast or outdrink Banach during these sessions. We discussed problems proposed right there, often with no solution evident even after several hours of thinking. The next day Banach was likely to appear with several small sheets of paper containing outlines of proofs he had completed.

Andrzej Turowicz, also a professor of mathematics at the an Kazimierz University in Lvov, also described Banach's style of working (see [3]):-

[Banach] would spend most of his days in cafés, not only in the company of others but also by himself. He liked the noise and the music. They did not prevent him from concentrating and thinking. There were cases when, after the cafés closed for the night, he would walk over to the railway station where the cafeteria was open around the clock. There, over a glass of beer, he would think about his problems.

In 1939, just before the start of World War II, Banach was elected as President of the Polish Mathematical Society. At the beginning of the war Soviet troops occupied Lvov. Banach had been on good terms with the Soviet mathematicians before the war started, visiting Moscow several times, and he was treated well by the new Soviet administration. He was allowed to continue to hold his chair at the university and he became the Dean of the Faculty of Science at the university, now renamed the Ivan Franko University. Banach's father came to Lvov fleeing from the German armies advancing towards Kraków. Life at this stage was little changed for Banach who continued his research, his textbook writing, lecturing and sessions in the cafés. Sobolev and Aleksandrov visited Banach in Lvov in 1940, while Banach attended conferences in the Soviet Union. He was in Kiev when Germany invaded the Soviet Union and he returned immediately to his family in Lvov.

The Nazi occupation of Lvov in June 1941 meant that Banach lived under very difficult conditions. He was arrested under suspicion of trafficking in German currency but released after a few weeks. He survived a period when Polish academics were murdered, his doctoral supervisor Lomnicki dying on the tragic night of 3 July 1941 when many massacres occurred. Towards the end of 1941 Banach worked feeding lice in German institute dealing with infectious diseases. Feeding lice was to be his life during the remainder of the Nazi occupation of Lvov up to July 1944. As soon as the Soviet troops retook Lvov Banach renewed his contacts. He met Sobolev outside Moscow but clearly he was by this time seriously ill. Sobolev, giving an address at a memorial conference for Banach, said of this meeting (see for example [3]):-

Despite heavy traces of the war years under German occupation, and despite the grave illness that was undercutting his strength, Banach's eyes were still lively. He remained the same sociable, cheerful, and extraordinarily well-meaning and charming Stefan Banach whom I had seen in Lvov before the war. That is how he remains in my memory: with a great sense of humour, an energetic human being, a beautiful soul, and a great talent.

Banach planned to go to Kraków after the war to take up the chair of mathematics at the Jagiellonian University but he died in Lvov in 1945 of lung cancer.

Banach founded modern functional analysis and made major contributions to the theory of topological vector spaces. In addition, he contributed to measure theory, integration, the theory of sets, and orthogonal series.

In his dissertation, written in 1920, he defined axiomatically what today is called a Banach space. The idea was introduced by others at about the same time, for example Wiener introduced the notion but did not develop the theory. The name 'Banach space' was coined by Fréchet. Banach algebras were also named after him.

A Banach space is a real or complex normed vector space that is complete as a metric space under the metric

d(x, y) = ||x-y||

induced by the norm. The completeness is important as this means that Cauchy sequences in Banach spaces converge.

A Banach algebra is a Banach space where the norm satisfies

||xy|| ||x||.||y||

The importance of Banach's contribution is that he developed a systematic theory of functional analysis, where before there had only been isolated results which were later seen to fit into the new theory. The theory generalised the contributions made by Volterra, Fredholm and Hilbert on integral equations.

Banach proved a number of fundamental results on normed linear spaces, and many important theorems are today named after him. There is the Hahn-Banach theorem on the extension of continuous linear functionals, the Banach-Steinhaus theorem on bounded families of mappings, the Banach-Alaoglu theorem, the Banach fixed point theorem and the Banach-Tarski paradoxical decomposition of a ball.

The Banach-Tarski paradox appeared in a joint paper of the two mathematicians in 1926 in Fundamenta Mathematicae entitled Sur la décomposition des ensembles de points en partiens respectivement congruent. The puzzling paradox shows that a ball can be divided up into subsets which can be fitted together to make two balls each identical to the first. The axiom of choice is needed to define the decomposition and the fact that it is able to give such a non-intuitive result has made some mathematicians question the use of the axiom. The Banach-Tarski paradox was a major contribution to the work being done on axiomatic set theory around this period.

Banach's open mapping of 1929 also uses set-theoretic concepts, this time concepts introduced by Baire in his 1899 dissertation.

Abu Ja'far Muhammad ibn Musa Al-Khwarizmi

Born: about 780 in Baghdad (now in Iraq)

Died: about 850

We know few details of Abu Ja'far Muhammad ibn Musa al-Khwarizmi's life. One unfortunate effect of this lack of knowledge seems to be the temptation to make guesses based on very little evidence. In [1] Toomer suggests that the name al-Khwarizmi may indicate that he came from Khwarizm south of the Aral Sea in central Asia. He then writes:-

But the historian al-Tabari gives him the additional epithet "al-Qutrubbulli", indicating that he came from Qutrubbull, a district between the Tigris and Euphrates not far from Baghdad, so perhaps his ancestors, rather than he himself, came from Khwarizm ... Another epithet given to him by al-Tabari, "al-Majusi", would seem to indicate that he was an adherent of the old Zoroastrian religion. ... the pious preface to al-Khwarizmi's "Algebra" shows that he was an orthodox Muslim, so Al-Tabari's epithet could mean no more than that his forebears, and perhaps he in his youth, had been Zoroastrians.

However, Rashed [7], put a rather different interpretation on the same words by Al-Tabari:-

... Al-Tabari's words should read: "Muhammad ibn Musa al-Khwarizmi and al-Majusi al-Qutrubbulli ...", (and that there are two people al-Khwarizmi and al-Majusi al-Qutrubbulli): the letter "wa" was omitted in the early copy. This would not be worth mentioning if a series of conclusions about al-Khwarizmi's personality, occasionally even the origins of his knowledge, had not been drawn. In his article ([1]) G J Toomer, with naive confidence, constructed an entire fantasy on the error which cannot be denied the merit of making amusing reading.

This is not the last disagreement that we shall meet in describing the life and work of al-Khwarizmi. However before we look at the few facts about his life that are known for certain, we should take a moment to set the scene for the cultural and scientific background in which al-Khwarizmi worked.

Harun al-Rashid became the fifth Caliph of the Abbasid dynasty on 14 September 786, about the time that al-Khwarizmi was born. Harun ruled, from his court in the capital city of Baghdad, over the Islam empire which stretched from the Mediterranean to India. He brought culture to his court and tried to establish the intellectual disciplines which at that time were not flourishing in the Arabic world. He had two sons, the eldest was al-Amin while the younger was al-Mamun. Harun died in 809 and there was an armed conflict between the brothers.

Al-Mamun won the armed struggle and al-Amin was defeated and killed in 813. Following this, al-Mamun became Caliph and ruled the empire from Baghdad. He continued the patronage of learning started by his father and founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated. He also built up a library of manuscripts, the first major library to be set up since that at Alexandria, collecting important works from Byzantium. In addition to the House of Wisdom, al-Mamun set up observatories in which Muslim astronomers could build on the knowledge acquired by earlier peoples.

Al-Khwarizmi and his colleagues the Banu Musa were scholars at the House of Wisdom in Baghdad. Their tasks there involved the translation of Greek scientific manuscripts and they also studied, and wrote on, algebra, geometry and astronomy. Certainly al-Khwarizmi worked under the patronage of Al-Mamun and he dedicated two of his texts to the Caliph. These were his treatise on algebra and his treatise on astronomy. The algebra treatise Hisab al-jabr w'al-muqabala was the most famous and important of all of al-Khwarizmi's works. It is the title of this text that gives us the word "algebra" and, in a sense that we shall investigate more fully below, it is the first book to be written on algebra.

Rosen's translation of al-Khwarizmi's own words describing the purpose of the book tells us that al-Khwarizmi intended to teach [11] (see also [1]):-

... what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.

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 [11]:-

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 [11]:-

... 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 cross ⁵/₂ = ²⁵/₄. Hence the outside square in Fig 3 has area 4 cross 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.

These geometrical proofs are a matter of disagreement between experts. The question, which seems not to have an easy answer, is whether al-Khwarizmi was familiar with Euclid's Elements. We know that he could have been, perhaps it is even fair to say "should have been", familiar with Euclid's work. In al-Rashid's reign, while al-Khwarizmi was still young, al-Hajjaj had translated Euclid's Elements into Arabic and al-Hajjaj was one of al-Khwarizmi's colleagues in the House of Wisdom. This would support Toomer's comments in [1]:-

... in his introductory section al-Khwarizmi uses geometrical figures to explain equations, which surely argues for a familiarity with Book II of Euclid's "Elements".

Rashed [9] writes that al-Khwarizmi's:-

... treatment was very probably inspired by recent knowledge of the "Elements".

However, Gandz in [6] (see also [23]), argues for a very different view:-

Euclid's "Elements" in their spirit and letter are entirely unknown to [al-Khwarizmi]. Al-Khwarizmi has neither definitions, nor axioms, nor postulates, nor any demonstration of the Euclidean kind.

I [EFR] think that it is clear that whether or not al-Khwarizmi had studied Euclid's Elements, he was influenced by other geometrical works. As Parshall writes in [35]:-

... because his treatment of practical geometry so closely followed that of the Hebrew text, Mishnat ha Middot, which dated from around 150 AD, the evidence of Semitic ancestry exists.

Al-Khwarizmi continues his study of algebra in Hisab al-jabr w'al-muqabala by examining how the laws of arithmetic extend to an arithmetic for his algebraic objects. For example he shows how to multiply out expressions such as

(a + b x) (c + d x)

although again we should emphasise that al-Khwarizmi uses only words to describe his expressions, and no symbols are used. Rashed [9] sees a remarkable depth and novelty in these calculations by al-Khwarizmi which appear to us, when examined from a modern perspective, as relatively elementary. He writes [9]:-

Al-Khwarizmi's concept of algebra can now be grasped with greater precision: it concerns the theory of linear and quadratic equations with a single unknown, and the elementary arithmetic of relative binomials and trinomials. ... The solution had to be general and calculable at the same time and in a mathematical fashion, that is, geometrically founded. ... The restriction of degree, as well as that of the number of unsophisticated terms, is instantly explained. From its true emergence, algebra can be seen as a theory of equations solved by means of radicals, and of algebraic calculations on related expressions...

If this interpretation is correct, then al-Khwarizmi was as Sarton writes:-

... the greatest mathematician of the time, and if one takes all the circumstances into account, one of the greatest of all time....

In a similar vein Rashed writes [9]:-

It is impossible to overstress the originality of the conception and style of al-Khwarizmi's algebra...

but a different view is taken by Crossley who writes [4]:-

[Al-Khwarizmi] may not have been very original...

and Toomer who writes in [1]:-

... Al-Khwarizmi's scientific achievements were at best mediocre.

In [23] Gandz gives this opinion of al-Khwarizmi's algebra:-

Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers.

The next part of al-Khwarizmi's Algebra consists of applications and worked examples. He then goes on to look at rules for finding the area of figures such as the circle and also finding the volume of solids such as the sphere, cone, and pyramid. This section on mensuration certainly has more in common with Hindu and Hebrew texts than it does with any Greek work. The final part of the book deals with the complicated Islamic rules for inheritance but require little from the earlier algebra beyond solving linear equations.

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 [19]) 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 [1]:-

... 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 [17].

Another important work by al-Khwarizmi was his work Sindhind zij on astronomy. The work, described in detail in [48], is based in Indian astronomical works [47]:-

... 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. In the tenth century al-Majriti made a critical revision of the shorter version and this was translated into Latin by Adelard of Bath. 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. A related manuscript, attributed to al-Khwarizmi, on spherical trigonometry is discussed in [39].

Although his astronomical work is based on that of the Indians, and most of the values from which he constructed his tables came from Hindu astronomers, al-Khwarizmi must have been influenced by Ptolemy's work too [1]:-

It is certain that Ptolemy's tables, in their revision by Theon of Alexandria, were already known to some Islamic astronomers; and it is highly likely that they influenced, directly or through intermediaries, the form in which Al-Khwarizmi's tables were cast.

Al-Khwarizmi wrote a major work on geography which give latitudes and longitudes for 2402 localities as a basis for a world map. The book, which is based on Ptolemy's Geography, lists with latitudes and longitudes, cities, mountains, seas, islands, geographical regions, and rivers. The manuscript does include maps which on the whole are more accurate than those of Ptolemy. In particular it is clear that where more local knowledge was available to al-Khwarizmi such as the regions of Islam, Africa and the Far East then his work is considerably more accurate than that of Ptolemy, but for Europe al-Khwarizmi seems to have used Ptolemy's data.

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 a quote by Mohammad Kahn, given in [3]:-

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...

The teaching of mathematics: Some conclusions

an article by J J O'Connor and E F Robertson

The fortunes of Mathematics in Education have varied considerably through the ages, from the highest respect and devotion in Greece, its almost disappearance in the Mediaeval ages, to its subsequent re-emergence in the modern times. The key changes in this development have been in response to a small number of events in history and the actions of a few people and organisations.

These can be summarised as follows:

1. The fall of the Roman Empire and the subsequent loss of knowledge and educational practises due to the succession of wars that followed this event.

2. The efforts of a few key people, Charlemagne and Alcuin being probably the foremost among them, to improve educational standards and the knowledge of the general populace and the clergy. Pope Sylvester II also played his part in improving the Church's opinion of Mathematics in the later period of the Dark Ages.

3. The increase in knowledge thanks to texts saved and recovered by the Arabs. Brought to Europe by knights on crusades, and the work of Fibonacci in introducing and promoting the new and improved numerical systems.

4. The rise in commerce and navigation during the Renaissance which meant that people with a good level of mathematical knowledge were sought after as tutors for individuals, or teachers for schools of trade and navigation that were beginning to appear.

5. The invention of the printing press which led to a much wider dissemination of knowledge and mathematical advances thanks to the reduced cost of buying or acquiring books and texts.

6. The foundation of further universities as centres of knowledge and learning.

7. The effect of the Reformation of the Church in both Scotland and England had far reaching consequences for educational standards. Scotland experienced a rise in both the number of schools, and the quality of education supplied by them, and England saw the added effect of the Act of Uniformity in the establishment of the Dissident Academies, many of whom were more open to the mathematical sciences then the traditional Grammar Schools and Universities. Scottish Councils copied this with the foundation of several mathematically strong Academies in Perth, Dundee and other cities.

8. Finally the effect of the industrial revolution with the increased numbers of immigrant workers from the rural areas which because of the rise in illiteracy and lack of numerate skills highlighted the lack of education available there and the insufficient services in the cities.

All of these factors and events influenced the position of Mathematics in society and education, and the opinions of the public to the subject. The struggle to highlight the importance of a sound mathematical understanding needed in today's world continues with efforts aimed at improving the image of the subject being sponsored and run by both governments and public organisations.

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