Abu Abd Allah Muhammad ibn Isa Al-Mahani

Abu Abd Allah Muhammad ibn Isa Al-MahaniBorn: about 820 in Mahan, Kerman, Persia (now Iran)Died: 880 in Baghdad, IraqThere is very little information regarding al-Mahani's life. We do know a little about al-Mahani's work in astronomy from Ibn Yunus's astronomical handbook al-Zij al-Hakimi al-kabir. In this work Ibn Yunus quotes from writings by al-Mahani, which have since been lost, which describe observations which al-Mahani made between the years 853 and 866. At least we have accurate dating of al-Mahani's life from this source. Ibn Yunus writes that al-Mahani observed lunar eclipses and [1]:- ... he calculated their beginnings with an astrolabe and that the beginnings of three consecutive eclipses were about half an hour later than calculated. The Fihrist (Index) was a work compiled by the bookseller Ibn an-Nadim in 988. It gives a full account of the Arabic literature which was available in the 10th century and in particular mentions al-Mahani, not for his work in astronomy, but rather for his work in geometry and arithmetic. However the work which al-Mahani did in mathematics may well have been motivated by various problems of an astronomical nature. We know that some of al-Mahani's work in algebra was motivated by trying to solve problems due to Archimedes. The problem of Archimedes which he attempted to solve in a novel way was that of cutting a sphere by a plane so that the two resulting segments had volumes of a given ratio. It was Omar Khayyam, giving an important historical description of algebra, who puts al-Mahani's work into context. Omar Khayyam writes (see for example [2] or [3]):- Al-Mahani was one of the modern authors who conceived the idea of solving the auxiliary theorem used by Archimedes in the fourth proposition of the second book of his treatise on the sphere and the cylinder algebraically. However, he was led to an equation involving cubes, squares and numbers which he failed to solve after giving it lengthy meditation. Therefore, this solution was declared impossible until the appearance of Ja'far al-Khazin who solved the equation with the help of conic sections. Omar Khayyam is quite correct to rate this work highly. It would be too easy to say that since al-Mahani has proposed a method of solution which he could not carry through then his work was of little value. However, this, as Omar Khayyam is well aware, is not so at all and the fact that al-Mahani conceived the idea of reducing problems such as duplicating the cube to problems in algebra was an important step forward. A number of works by al-Mahani have survived, in particular commentaries which he wrote on parts of Euclid's Elements. In particular his work on ratio and irrational ratios which are contained in commentaries he gave on Books V and X of the Elements survive as does his attempt to clarify difficult parts of Book XIII. He also wrote a work which gives those 26 propositions in Book I which can be proved without using a reductio ad absurdum argument but this work has been lost. Also lost is his work attempting to improve the descriptions given by Menelaus in his Spherics. Article by: J J O'Connor and E F Robertson November 1999 MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Mahani.html]

Abu Sahl Waijan ibn Rustam al-Quhi

Abu Sahl Waijan ibn Rustam al-QuhiBorn: about 940 in Tabaristan (now Mazanderan), Persia (now Iran)Died: about 1000There are two spellings of al-Quhi's name in English which seem to appear about equally often, namely al-Quhi and al-Kuhi. We can deduce from al-Quhi's name that he came from the village of Quh in Tabaristan. He was brought up during the period that a new dynasty was being established which would rule over Iran. The Buyid Islamic dynasty ruled in western Iran and Iraq from 945 to 1055 in the period between the Arab and Turkish conquests. The period began in 945 when Ahmad Buyeh occupied the 'Abbasid capital of Baghdad. The high point of the Buyid dynasty was during the reign of 'Adud ad-Dawlah from 949 to 983. He ruled from Baghdad over all southern Iran and most of what is now Iraq. A great patron of science and the arts, 'Adud ad-Dawlah supported a number of mathematicians at his court in Baghdad, including al-Quhi, Abu'l-Wafa and al-Sijzi. In 969 'Adud ad-Dawlah ordered that observations be made of the winter and summer solstices in Shiraz. These observations of the winter and summer solstices were made by al-Quhi, al-Sijzi and other scientists in Shiraz during 969/970. Sharaf ad-Dawlah was 'Adud ad-Dawlah's son and he became Caliph in 983. He continued to support mathematics and astronomy so al-Quhi remained at the court in Baghdad working for the new Caliph. Sharaf ad-Dawlah required al-Quhi to make observations of the seven planets and in order to do this al-Quhi had an observatory built in the garden of the palace in Baghdad. The instruments in the observatory were built to al-Quhi's own design and installed once the building was complete. Al-Quhi was made director of the observatory and it was officially opened in June 988. A number of scientists were present at the opening. One in particular, the famous mathematician and astronomer Abu'l-Wafa, is worthy of mention. He was also employed at the court of Sharaf ad-Dawlah. Another who was present at the opening was Abu Ishaq al-Sabi. Al-Sabi was a high ranking official in Baghdad who was interested in mathematics. We mention later in this article correspondence between al-Quhi and al-Sabi. Some accurate observations were made but the observatory ceased work in 989 on the death of Sharaf ad-Dawlah. The Buyid dynasty was by this stage beginning to lose control of the empire. The economy was on a downward path, and rebellions in the army made the ruler's life difficult. Fine cultural activities such as an observatory took a lower priority. Our description of al-Quhi's life has highlighted his work in astronomy. However, it is in mathematics that he is more famous, being the leading figure in a revival and continuation of Greek higher geometry in the Islamic world. The geometric problems that al-Quhi studied usually led to quadratic or cubic equations. Nasir al-Din al-Tusi described one of the problems considered by al-Quhi writing (see for example [1]):- To construct a sphere segment equal in volume to a given sphere segment and equal in area to a second sphere segment - a problem similar to but more difficult than related problems solved by Archimedes - Al-Quhi constructed the two unknown lengths by intersecting an equilateral hyperbola with a parabola and rigorously discussed the conditions under which the problem is soluble. Al-Quhi's solution to the problem is given in [5]. It is a classical style of solution using results from Euclid's Elements, Apollonius's Conics and Archimedes' On the sphere and cylinder. If a solution exists, al-Quhi showed that it will have coordinates which lie on a particular rectangular hyperbola that he has constructed. Of course, al-Quhi does not express the mathematics in these modern terms but rather in the usual classical geometry of ancient Greek mathematics. Next al-Quhi introduces the "cone of the surface" which, after many deductions, leads to showing that the solution has coordinates lying on a parabola. The problem is then beautifully solved as the intersection of the two curves. In another treatise On the construction of an equilateral pentagon in a known square al-Quhi solves the problem given in the title again using the intersection of two conic sections, this time two hyperbolas. Although it is impossible to inscribe a regular pentagon in a square, an equilateral pentagon can be inscribed in two ways. One, which requires the solution of a quadratic equation, had been found by Abu Kamil in the ninth century. The other, which requires the solution of a quartic equation, is the one presented by al-Quhi. Details of this treatise are given in [6] (see the corrections and additions of [7]), and [8]. Al-Quhi also described a conic compass, a compass with one leg of variable length, for drawing conic sections in the treatise On the perfect compass [1]:- ... he first described the method of constructing straight lines, circles, and conic sections with this compass, and then treated the theory. He concluded that one could now easily construct astrolabes, sundials and similar instruments. Indeed al-Quhi did consider the problem of constructing astrolabes in On the construction of the astrolabe. The astrolabe was an instrument used to observe altitudes, and it provided a mechanical means to transform celestial coordinates between an equatorial system and one based on the horizon. This treatise is in two Books, the first being divided into four chapters, the second book into seven chapters. There are a number of difficult mapping problems solved by al-Quhi in this work. In particular, using a method resembling descriptive geometry, he maps circles on the sphere into the equatorial plane. After manipulation, they are mapped back again onto the sphere in a remarkable piece of visualisation. Despite the appearance of the work being of practical use in constructing an astrolabe, it would appear that al-Quhi was more interested in the mathematics for its own sake than he was in giving a practical manual. Finally we should mention the correspondence between al-Quhi and al-Sabi which we mentioned above. It is known that there were at least six letters exchanged but only details of four survive. These are given in both Arabic and English in [3]. Topics covered are quite varied, ranging from a discussion of what "known" means to solutions of specific problems such as the following Suppose we are given a circle and two intersecting straight lines l and m. Suppose the tangent to the circle at a point T meets l at L and m at M. How can one choose T so that TL : TM is equal to a given ratio? Perhaps the most interesting parts of the correspondence are six theorems given by al-Quhi concerning the centres of gravity of various figures. Five of the six results are correct but the sixth is false. It states that the centre of gravity of a semicircle divides the radius in the ratio 3 : 7. From this false result al-Quhi deduces the equally false result that ? = 28/9. Even the best mathematicians can make mistakes! Article by: J J O'Connor and E F Robertson November 1999 MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Quhi.html]

Abu Yusuf Yaqub ibn Ishaq al-Sabbah Al-Kindi

Abu Yusuf Yaqub ibn Ishaq al-Sabbah Al-KindiBorn: about 801 in Kufah, IraqDied: 873 in Baghdad, IraqAl-Kindi was born and brought up in Kufah, which was a centre for Arab culture and learning in the 9th century. This was certainly the right place for al-Kindi to get the best education possible at this time. Although quite a few details (and legends) of al-Kindi's life are given in various sources, these are not all consistent. We shall try to give below details which are fairly well substantiated. According to [3], al-Kindi's father was the governor of Kufah, as his grandfather had been before him. Certainly all agree that al-Kindi was descended from the Royal Kindah tribe which had originated in southern Arabia. This tribe had united a number of tribes and reached a position of prominence in the 5th and 6th centuries but then lost power from the middle of the 6th century. However, descendants of the Royal Kindah continued to hold prominent court positions in Muslim times. After beginning his education in Kufah, al-Kindi moved to Baghdad to complete his studies and there he quickly achieved fame for his scholarship. He came to the attention of the Caliph al-Ma'mun who was at that time setting up the "House of Wisdom" in Baghdad. Al-Ma'mun had won an armed struggle against his brother in 813 and became Caliph in that year. He ruled his empire, first from Merv then, after 818, he ruled from Baghdad where he had to go to put down an attempted coup. Al-Ma'mun was a patron of learning and founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated. Al-Kindi was appointed by al-Ma'mun to the House of Wisdom together with al-Khwarizmi and the Banu Musa brothers. The main task that al-Kindi and his colleagues undertook in the House of Wisdom involved the translation of Greek scientific manuscripts. Al-Ma'mun had 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-Ma'mun set up observatories in which Muslim astronomers could build on the knowledge acquired by earlier peoples. In 833 al-Ma'mun died and was succeeded by his brother al-Mu'tasim. Al-Kindi continued to be in favour and al-Mu'tasim employed al-Kindi to tutor his son Ahmad. Al-Mu'tasim died in 842 and was succeeded by al-Wathiq who, in turn, was succeeded as Caliph in 847 by al-Mutawakkil. Under both these Caliphs al-Kindi fared less well. It is not entirely clear whether this was because of his religious views or because of internal arguments and rivalry between the scholars in the House of Wisdom. Certainly al-Mutawakkil persecuted all non-orthodox and non-Muslim groups while he had synagogues and churches in Baghdad destroyed. However, al-Kindi's [6]:- ... lack of interest in religious argument can be seen in the topics on which he wrote. ... he appears to coexist with the world view of orthodox Islam. In fact most of al-Kindi's philosophical writings seem designed to show that he believed that the pursuit of philosophy is compatible with orthodox Islam. This would seem to indicate that it is more probably that al-Kindi became [1]:- ... the victim of such rivals as the mathematicians Banu Musa and the astrologer Abu Ma'shar. It is claimed that the Banu Musa brothers caused al-Kindi to lose favour with al-Mutawakkil to the extent that he had him beaten and gave al-Kindi's library to the Banu Musa brothers. Al-Kindi was best known as a philosopher but he was also a mathematician and scientist of importance [3]:- To his people he became known as ... the philosopher of the Arabs. He was the only notable philosopher of pure Arabian blood and the first one in Islam. Al-Kindi "was the most leaned of his age, unique among his contemporaries in the knowledge of the totality of ancient scientists, embracing logic, philosophy, geometry, mathematics, music and astrology. Perhaps, rather surprisingly for a man of such learning whose was employed to translate Greek texts, al-Kindi does not appear to have been fluent enough in Greek to do the translation himself. Rather he polished the translations made by others and wrote commentaries on many Greek works. Clearly he was most influenced most strongly by the writings of Aristotle but the influence of Plato, Porphyry and Proclus can also be seen in al-Kindi's ideas. We should certainly not give the impression that al-Kindi merely borrowed from these earlier writer, for he built their ideas into an overall scheme which was certainly his own invention. Al-Kindi wrote many works on arithmetic which included manuscripts on Indian numbers, the harmony of numbers, lines and multiplication with numbers, relative quantities, measuring proportion and time, and numerical procedures and cancellation. He also wrote on space and time, both of which he believed were finite, 'proving' his assertion with a paradox of the infinite. Garro gives al-Kindi's 'proof' that the existence of an actual infinite body or magnitude leads to a contradiction in [7]. In his more recent paper [8], Garro formulates the informal axiomatics of al-Kindi's paradox of the infinite in modern terms and discusses the paradox both from a mathematical and philosophical point of view. In geometry al-Kindi wrote, among other works, on the theory of parallels. He gave a lemma investigating the possibility of exhibiting pairs of lines in the plane which are simultaneously non-parallel and non-intersecting. Also related to geometry was the two works he wrote on optics, although he followed the usual practice of the time and confused the theory of light and the theory of vision. Perhaps al-Kindi's own words give the best indication of what he attempted to do in all his work. In the introduction to one of his books he wrote (see for example [1]):- It is good ... that we endeavour in this book, as is our habit in all subjects, to recall that concerning which the Ancients have said everything in the past, that is the easiest and shortest to adopt for those who follow them, and to go further in those areas where they have not said everything ... Certainly al-Kindi tried hard to follow this path. For example in his work on optics he is critical of a Greek description by Anthemius of how a mirror was used to set a ship on fire during a battle. Al-Kindi adopts a more scientific approach (see for example [1]):- Anthemius should not have accepted information without proof ... He tells us how to construct a mirror from which twenty-four rays are reflected on a single point, without showing how to establish the point where the rays unite at a given distance from the middle of the mirror's surface. We, on the other hand, have described this with as much evidence as our ability permits, furnishing what was missing, for he has not mentioned a definite distance. Much of al-Kindi's work remains to be studied closely or has only recently been subjected to scholarly research. For example al-Kindi's commentary on Archimedes' The measurement of the circle has only received careful attention as recently as the 1993 publication [10] by Rashed. Article by: J J O'Connor and E F Robertson November 1999 MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Kindi.html]

Abu Mahmud Hamid ibn al-Khidr Al-Khujandi

Abu Mahmud Hamid ibn al-Khidr Al-KhujandiBorn: about 940 in Khudzhand, TajikistanDied: 1000Few facts about al-Khujandi's life are known. What little we know comes through his writings which have survived and also some comments made by Nasir al-din al-Tusi. From al-Tusi's comments we can be fairly certain that al-Khujandi came from the city of Khudzhand. The city lies along both banks of the Syrdarya river, at the entrance to the fertile Fergana Valley, and it was captured by the Arabs in the 8th century. Al-Tusi says that al-Khujandi was one of the rulers of the Mongol tribe in that region so he must have come from the nobility. Al-Khujandi was supported in his scientific work for most of his life by members of the Buyid dynasty. The dynasty came to power in 945 when Ahmad ad-Dawlah occupied the 'Abbasid capital of Baghdad. Members of Ahmad ad-Dawlah's family became rulers in different provinces and there was never a great deal of cohesion in the Buyid empire. Al-Khujandi received patronage from Fakhr ad-Dawlah who ruled from 976 to 997. It was Fakhr ad-Dawlah who supported al-Khujandi in his major project to construct a huge mural sextant for his observatory at Rayy, which is near modern Tehran. It was believed by many Arabic scientists that the larger an instrument was, the more accurate were the results obtained. In fact al-Khujandi's mural sextant was his own invention and it did break new ground in having a scale which indicated seconds, a level of accuracy never before attempted. During the year 994 al-Khujandi used the very large instrument to observe a series of meridian transits of the sun near the solstices. He used these observations, made on 16 and 17 June 994 for the summer solstice and 14 and 17 December 994 for the winter solstice, to calculate the obliquity of the ecliptic, and the latitude of Rayy. He described his measurements in detail in a treatise On the obliquity of the ecliptic and the latitudes of the cities. From his observations he obtained 23 32' 19" for the obliquity of the ecliptic. This value was lower than values obtained previously [1]:- Al-Khujandi says that the Indians found the greatest obliquity of the ecliptic, 24 ; Ptolemy 23 51' ; himself 23 32' 19". These divergent values cannot be due to defective instruments. Actually the obliquity of the ecliptic is not constant; it is a decreasing quantity. There is, however, an error in al-Khujandi's value for the obliquity of the ecliptic; it is about two minutes too low. The error was discussed by al-Biruni in his Tahdid where he claimed that the aperture of the sextant settled about one span in the course of al-Khujandi's observations due to the weight of the instrument. Al-Biruni is almost certainly correct in pinpointing the cause of the error. However, al-Khujandi's latitude for Rayy, 35 34' 38.45", despite being calculated using his erroneous value for the obliquity of the ecliptic, is accurate to the nearest minute of arc. It remains for us to discuss the claim that al-Khujandi discovered the sine theorem. The claim was made by al-Tusi who gives al-Khujandi's proof of the result for spherical triangles in his Shakl al-qatta. Although there is no reason to doubt al-Tusi that the proof he gives does indeed come from al-Khujandi there is quite a few reason to believe that one of Abu'l-Wafa or Abu Nasr Mansur was the original discoverer. Both Abu'l-Wafa and Abu Nasr Mansur claim to have discovered the sine theorem while, as far as we are aware, al-Khujandi makes no such claim. Also al-Khujandi was more of a designer of astronomical instruments and an astronomical observer than he was theoretician. Finally, although this really proves little, the theorem appears many times in the writings of Abu Nasr Mansur: both his writings on geometry as well as those on astronomy. We should make one final comment on the mathematical contributions of al-Khujandi. He stated Fermat's Last Theorem in the case n = 3 although, not surprisingly, his proof is wrong. Al-Khazin wrote:- I demonstrated earlier ... that what Abu Muhammad al-Khujandi advanced - may God have mercy on him - in his demonstration that the sum of two cubic numbers is not a cube, is defective and incorrect. It is certainly interesting that al-Khujandi, despite his practical rather than theoretical achievements, should be interested in this number theory result. Article by: J J O'Connor and E F Robertson November 1999 MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Khujandi.html]

Omar Khayyam

Omar KhayyamBorn: 18 May 1048 in Nishapur, Persia (now Iran)Died: 4 Dec 1131 in Nishapur, Persia (now Iran)Omar Khayyam's full name was Ghiyath al-Din Abu'l-Fath Umar ibn Ibrahim Al-Nisaburi al-Khayyami. A literal translation of the name al-Khayyami (or al-Khayyam) means 'tent maker' and this may have been the trade of Ibrahim his father. Khayyam played on the meaning of his own name when he wrote:- Khayyam, who stitched the tents of science,Has fallen in grief's furnace and been suddenly burned,The shears of Fate have cut the tent ropes of his life,And the broker of Hope has sold him for nothing! The political events of the 11th Century played a major role in the course of Khayyam's life. The Seljuq Turks were tribes that invaded southwestern Asia in the 11th Century and eventually founded an empire that included Mesopotamia, Syria, Palestine, and most of Iran. The Seljuq occupied the grazing grounds of Khorasan and then, between 1038 and 1040, they conquered all of north-eastern Iran. The Seljuq ruler Toghrïl Beg proclaimed himself sultan at Nishapur in 1038 and entered Baghdad in 1055. It was in this difficult unstable military empire, which also had religious problems as it attempted to establish an orthodox Muslim state, that Khayyam grew up. Khayyam studied philosophy at Naishapur and one of his fellow students wrote that he was:- ... endowed with sharpness of wit and the highest natural powers ... However, this was not an empire in which those of learning, even those as learned as Khayyam, found life easy unless they had the support of a ruler at one of the many courts. Even such patronage would not provide too much stability since local politics and the fortunes of the local military regime decided who at any one time held power. Khayyam himself described the difficulties for men of learning during this period in the introduction to his Treatise on Demonstration of Problems of Algebra (see for example [1]):- I was unable to devote myself to the learning of this algebra and the continued concentration upon it, because of obstacles in the vagaries of time which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes; and if they see a certain person seeking for the right and preferring the truth, doing his best to refute the false and untrue and leaving aside hypocrisy and deceit, they make a fool of him and mock him. However Khayyam was an outstanding mathematician and astronomer and, despite the difficulties which he described in this quote, he did write several works including Problems of Arithmetic, a book on music and one on algebra before he was 25 years old. In 1070 he moved to Samarkand in Uzbekistan which is one of the oldest cities of Central Asia. There Khayyam was supported by Abu Tahir, a prominent jurist of Samarkand, and this allowed him to write his most famous algebra work, Treatise on Demonstration of Problems of Algebra from which we gave the quote above. We shall describe the mathematical contents of this work later in this biography. Toghril Beg, the founder of the Seljuq dynasty, had made Esfahan the capital of his domains and his grandson Malik-Shah was the ruler of that city from 1073. An invitation was sent to Khayyam from Malik-Shah and from his vizier Nizam al-Mulk asking Khayyam to go to Esfahan to set up an Observatory there. Other leading astronomers were also brought to the Observatory in Esfahan and for 18 years Khayyam led the scientists and produced work of outstanding quality. It was a period of peace during which the political situation allowed Khayyam the opportunity to devote himself entirely to his scholarly work. During this time Khayyam led work on compiling astronomical tables and he also contributed to calendar reform in 1079. Cowell quotes The Calcutta Review No 59:- When the Malik Shah determined to reform the calendar, Omar was one of the eight learned men employed to do it, the result was the Jalali era (so called from Jalal-ud-din, one of the king's names) - 'a computation of time,' says Gibbon, 'which surpasses the Julian, and approaches the accuracy of the Gregorian style.' Khayyam measured the length of the year as 365.24219858156 days. Two comments on this result. Firstly it shows an incredible confidence to attempt to give the result to this degree of accuracy. We know now that the length of the year is changing in the sixth decimal place over a person's lifetime. Secondly it is outstandingly accurate. For comparison the length of the year at the end of the 19th century was 365.242196 days, while today it is 365.242190 days. In 1092 political events ended Khayyam's period of peaceful existence. Malik-Shah died in November of that year, a month after his vizier Nizam al-Mulk had been murdered on the road from Esfahan to Baghdad by the terrorist movement called the Assassins. Malik-Shah's second wife took over as ruler for two years but she had argued with Nizam al-Mulk so now those whom he had supported found that support withdrawn. Funding to run the Observatory ceased and Khayyam's calendar reform was put on hold. Khayyam also came under attack from the orthodox Muslims who felt that Khayyam's questioning mind did not conform to the faith. He wrote in his poem the Rubaiyat :- Indeed, the Idols I have loved so longHave done my Credit in Men's Eye much Wrong:Have drowned my Honour in a shallow cup,And sold my reputation for a Song. Despite being out of favour on all sides, Khayyam remained at the Court and tried to regain favour. He wrote a work in which he described former rulers in Iran as men of great honour who had supported public works, science and scholarship. Malik-Shah's third son Sanjar, who was governor of Khorasan, became the overall ruler of the Seljuq empire in 1118. Sometime after this Khayyam left Esfahan and travelled to Merv (now Mary, Turkmenistan) which Sanjar had made the capital of the Seljuq empire. Sanjar created a great centre of Islamic learning in Merv where Khayyam wrote further works on mathematics. The paper [18] by Khayyam is an early work on algebra written before his famous algebra text. In it he considers the problem:- Find a point on a quadrant of a circle in such manner that when a normal is dropped from the point to one of the bounding radii, the ratio of the normal's length to that of the radius equals the ratio of the segments determined by the foot of the normal. Khayyam shows that this problem is equivalent to solving a second problem:- Find a right triangle having the property that the hypotenuse equals the sum of one leg plus the altitude on the hypotenuse. This problem in turn led Khayyam to solve the cubic equation x3 + 200x = 20x2 + 2000 and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables. Perhaps even more remarkable is the fact that Khayyam states that the solution of this cubic requires the use of conic sections and that it cannot be solved by ruler and compass methods, a result which would not be proved for another 750 years. Khayyam also wrote that he hoped to give a full description of the solution of cubic equations in a later work [18]:- If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in this art will be prepared. Indeed Khayyam did produce such a work, the Treatise on Demonstration of Problems of Algebra which contained a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. In fact Khayyam gives an interesting historical account in which he claims that the Greeks had left nothing on the theory of cubic equations. Indeed, as Khayyam writes, the contributions by earlier writers such as al-Mahani and al-Khazin were to translate geometric problems into algebraic equations (something which was essentially impossible before the work of al-Khwarizmi). However, Khayyam himself seems to have been the first to conceive a general theory of cubic equations. Khayyam wrote (see for example [9] or [10]):- In the science of algebra one encounters problems dependent on certain types of extremely difficult preliminary theorems, whose solution was unsuccessful for most of those who attempted it. As for the Ancients, no work from them dealing with the subject has come down to us; perhaps after having looked for solutions and having examined them, they were unable to fathom their difficulties; or perhaps their investigations did not require such an examination; or finally, their works on this subject, if they existed, have not been translated into our language. Another achievement in the algebra text is Khayyam's realisation that a cubic equation can have more than one solution. He demonstrated the existence of equations having two solutions, but unfortunately he does not appear to have found that a cubic can have three solutions. He did hope that "arithmetic solutions" might be found one day when he wrote (see for example [1]):- Perhaps someone else who comes after us may find it out in the case, when there are not only the first three classes of known powers, namely the number, the thing and the square. The "someone else who comes after us" were in fact del Ferro, Tartaglia and Ferrari in the 16th century. Also in his algebra book, Khayyam refers to another work of his which is now lost. In the lost work Khayyam discusses the Pascal triangle but he was not the first to do so since al-Karaji discussed the Pascal triangle before this date. In fact we can be fairly sure that Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients. This follows from the following passage in his algebra book (see for example [1], [9] or [10]):- The Indians possess methods for finding the sides of squares and cubes based on such knowledge of the squares of nine figures, that is the square of 1, 2, 3, etc. and also the products formed by multiplying them by each other, i.e. the products of 2, 3 etc. I have composed a work to demonstrate the accuracy of these methods, and have proved that they do lead to the sought aim. I have moreover increased the species, that is I have shown how to find the sides of the square-square, quatro-cube, cubo-cube, etc. to any length, which has not been made before now. the proofs I gave on this occasion are only arithmetic proofs based on the arithmetical parts of Euclid's "Elements". In Commentaries on the difficult postulates of Euclid's book Khayyam made a contribution to non-euclidean geometry, although this was not his intention. In trying to prove the parallels postulate he accidentally proved properties of figures in non-euclidean geometries. Khayyam also gave important results on ratios in this book, extending Euclid's work to include the multiplication of ratios. The importance of Khayyam's contribution is that he examined both Euclid's definition of equality of ratios (which was that first proposed by Eudoxus) and the definition of equality of ratios as proposed by earlier Islamic mathematicians such as al-Mahani which was based on continued fractions. Khayyam proved that the two definitions are equivalent. He also posed the question of whether a ratio can be regarded as a number but leaves the question unanswered. Outside the world of mathematics, Khayyam is best known as a result of Edward Fitzgerald's popular translation in 1859 of nearly 600 short four line poems the Rubaiyat. Khayyam's fame as a poet has caused some to forget his scientific achievements which were much more substantial. Versions of the forms and verses used in the Rubaiyat existed in Persian literature before Khayyam, and only about 120 of the verses can be attributed to him with certainty. Of all the verses, the best known is the following:- The Moving Finger writes, and, having writ,Moves on: nor all thy Piety nor WitShall lure it back to cancel half a Line,Nor all thy Tears wash out a Word of it. Article by: J J O'Connor and E F Robertson July 1999 MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/Biographies/Khayyam.html]

Shams al-Din Abu Abdallah Al-Khalili

Shams al-Din Abu Abdallah Al-KhaliliBorn: about 1320 in possibly Damascus, SyriaDied: about 1380 in possibly Damascus, SyriaAl-Khalili's full name is Shams al-din Abu Abdallah Muhammad ibn Muhammad al-Khalili. As can be seen from the list of references, much of the study of the work of al-Khalili has been done by David A King, who also wrote the article in [1]. Note that the articles [3], [4] and [5] are reprinted in [2]. King writes:- Al-Khalili was an astronomer associated with the Umayyad Mosque in Damascus in the latter half of the fourteenth century, who compiled an extensive corpus of tables for timekeeping by the sun and regulating the astronomically defined time of Muslim prayer ... Of course, giving tables for timekeeping using astronomical events, requires a thorough understanding of geometry on the sphere and the work by al-Khalili can be seen as the end-product of the work of the Arabs on this mathematical topic. Of course, it is interesting to realise that Muslim mathematicians had to solve this type of problem for religious reasons, and the religious requirements made them delve much more deeply into this area of mathematics than was necessary to solve the much less critical problems of the agricultural calendar. The tables, which were not really studied by historians of mathematics until King worked on them in the 1970s, were used for many centuries in Damascus, Cairo and Istanbul. They consist of [1]:- ... tables for reckoning time by the sun, for the latitude of Damascus; tables for regulating the time of Muslim prayer, for the latitude of Damascus; tables of auxiliary mathematical functions for timekeeping by the sun for all latitudes; tables of auxiliary mathematical functions for solving the problems of spherical astronomy for all latitudes; a table for displaying ... the direction of Mecca, as a function of terrestrial latitude and longitude; and tables for converting lunar ecliptic coordinates to equatorial coordinates. Of course, al-Khalili did not do all this work without basing some of it on the work of earlier mathematicians, to see the magnitude of his task note that one table alone contains over 13000 entries. Tables for reckoning time by the sun and tables for regulating the time of Muslim prayer, computed for the latitude of Cairo, had been earlier computed by ibn Yunus. The astronomer al-Mizzi spent his early life in Egypt, then moved to Damascus where he converted ibn Yunus's table for use there. Al-Mizzi died around 1350 and the first two of al-Khalili's tables were improved versions of the ones produced by al-Mizzi, where al-Khalili had taken more accurate values for the terrestrial coordinates of Damascus. Al-Khalili's tables for solving the problems of spherical astronomy can be seen to be tables which solve spherical triangles using a method similar to the modern cosine rule. The tables are remarkable for their accuracy and Van Brummelen in [6] uses:- .... computer-based tests to determine, if possible, the methods of computation used by al-Khalili in the construction of his auxiliary tables. This paper suggests a possible interpolation scheme used by al-Khalili and shows up a deep understanding that al-Khalili must have had regarding errors in his calculations which show [6]:- ... a curious lack of concern for accuracy at an early stage of the calculation, followed by a careful computation at a later stage where the calculation is sensitive to error. The calculation of the direction of Mecca, as a function of terrestrial latitude and longitude, was one of the hardest of all problems of spherical trigonometry for which Islam required a solution. There is a puzzle which has not yet been explained. The tables produced by al-Khalili for the direction of Mecca must have been calculated using his own auxiliary tables (which would be the most accurate available). However, the tables giving the direction of Mecca are remarkable for their accuracy having errors of around 0.1. This is a greater degree of accuracy than would result if al-Khalili used his auxiliary tables in their present form. One possible solution is that al-Khalili had computed more accurate auxiliary tables before calculating his tables for the direction of Mecca but these are now lost. Article by: J J O'Connor and E F Robertson November 1999 MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Khalili.html]

Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji

Abu Bekr ibn Muhammad ibn al-Husayn Al-KarajiBorn: 13 April 953 in Baghdad (now in Iraq)Died: about 1029The first comment that we must make regards al-Karaji's name. It appears both as al-Karaji and as al-Karkhi but this is not a simple matter of two different transliterations of the same Arabic name. The significance is that Karaj is a city in Iran and if the mathematician's name is al-Karaji then certainly his family were from that city. On the other hand Karkh is one of the original suburbs of Baghdad which grew up outside the southern gate of the original city. The name al-Karkhi would indicate that the mathematician came from the suburb of Baghdad. Historians seem divided as to which of these interpretations is correct. The version al-Karkhi was proposed by Woepcke (see [7] or [8]) but al-Karaji, the version which is most often used in texts today, was suggested as most likely by della Vida in 1933. Rashed comments (see [1] or [5]):- In the present state of our knowledge delle Vida's argument is plausible but not decisive. On the basis of the manuscripts consulted it is far from easy to decide in favour of either name. Certainly we know that al-Karaji lived in Baghdad for most of his life and that his chief mathematical works were written during the time when he lived in that city. His important treatise on algebra Al-Fakhri was dedicated to the ruler of Baghdad and was written in the city. However, at some later point in his career, al-Karaji left Baghdad to live in what are described as the "mountain countries". He seems to have given up mathematics at this time and concentrated on engineering topics such as the drilling of wells. The importance of al-Karaji in the development of mathematics is viewed rather differently by different authors. The reason for this, rather in the same spirit as the different views on al-Khwarizmi, depends on the significance one attaches to the style of his mathematics. Some consider that his work is merely reworking ideas from earlier mathematicians while others see him as the first person to completely free algebra from geometrical operations and replace them with the arithmetical type of operations which are at the core of algebra today. Crossley [3] sounds relatively unimpressed by al-Karaji's contributions (although he describes the content accurately):- [Al-Karaji] gives rules for the arithmetic operations including (essentially) the multiplication of polynomials. ... al-Karaji usually gives a numerical example for his rules but does not give any sort of proof beyond giving geometrical pictures. Often he explicitely says that he is giving a solution in the style of Diophantus. He does not treat equations above the second degree except for ones which can easily be reduced to at most second degree equations followed by the extraction of roots. The solutions of quadratics are based explicitly on the Euclidean theorems ... Woepcke in [7] (see also the reprint [8]) was the first historian to realise the importance of al-Karaji's work and later historians mostly agree with his interpretation. He describes it as the first appearance of a:- ... theory of algebraic calculus ... . Rashed (see [5] which contains Rashed's article from [1] and other writings by Rashed on al-Karaji) agrees with Woepcke's interpretation and perhaps goes even further in stressing al-Karaji's importance. He writes:- ... the more-or-less explicit aim of [al-Karaji's] exposition was to find the means of realising the autonomy and specificity of algebra, so as to be in a position to reject, in particular, the geometric representation of algebraic operations. To give another quote from Rashed's description of al-Karaji's contribution:- Al-Karaji's work holds an especially important place in the history of mathematics. ... the discovery and reading of the arithmetical work of Diophantus, in the light of the algebraic conceptions and methods of al-Khwarizmi and other Arab algebraists, made possible a new departure in algebra by Al-Karaji ... So what was this new departure in algebra? Perhaps it is best described by al-Samawal, one of al-Karaji's successors, who described it as [5]:- ... operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known. What al-Karaji achieved in Al-Fakhri was first to define the monomials x, x2, x3, ... and 1/x, 1/x2, 1/x3, ... and to give rules for products of any two of these. So what he achieved here was defining the product of these terms without any reference to geometry. In fact he almost gave the formula xnxm = xm+n for all integers n and m but he failed to make the definition x0= 1 so he fell just a little short. Having given rules for multiplication and division of monomials al-Karaji then looked at "composite quantities" or sums of monomials. For these he gave rules for addition, subtraction and multiplication but not for division in the general case, only giving rules for the division of a composite quantity by a monomial. He was able to give a rule for finding the square root of a composite quantity which is not completely general since it required the coefficients to be positive, but it is still a remarkable achievement. Al-Karaji also uses a form of mathematical induction in his arguments, although he certainly does not give a rigorous exposition of the principle. Basically what al-Karaji does is to demonstrate an argument for n = 1, then prove the case n = 2 based on his result for n = 1, then prove the case n = 3 based on his result for n = 2, and carry on to around n = 5 before remarking that one can continue the process indefinitely. Although this is not induction proper, it is a major step towards understanding inductive proofs. One of the results on which al-Karaji uses this form of induction comes from his work on the binomial theorem, the binomial coefficients and the Pascal triangle. In Al-Fakhri al-Karaji computed (a+b)3 and in Al-Badi he computed (a-b)3 and (a+b)4. The general construction of the Pascal triangle was given by al-Karaji in work described in the later writings of al-Samawal. In the translation by Rashed and Ahmad (see for example [5]) al-Samawal writes:- Let us now recall a principle for knowing the necessary number of multiplications of these degrees by each other, for any number divided into two parts. Al-Karaji said that in order to succeed we must place 'one' on a table and 'one' below the first 'one', move the first 'one' into a second column, add the first 'one' to the 'one' below it. Thus we obtain 'two', which we put below the transferred 'one' and we place the second 'one' below the 'two'. We have therefore 'one', 'two', and 'one'. To see how the second column of 1,2,1 corresponds to squaring a+b al-Samawal continues to describe Al-Karaji's work writing:- This shows that for every number composed of two numbers, if we multiple each of them by itself once - since the two extremes are 'one' and 'one' - and if we multiply each one by the other twice - since the intermediate term is 'two' - we obtain the square of this number. This is a beautiful description of the binomial theorem using the Pascal triangle. The description continues up to the binomial coefficients which give (a+b)5 but we shall only quote how al-Karaji constructs the third column from the second:- If we transfer the 'one' in the second column into a third column, then add 'one' from the second column to 'two' below it, we obtain 'three' to be written under the 'one' in the third column. If we then add 'two' from the second column to ''one' below it we have 'three' which is written under the 'three', then we write 'one' under this 'three'; we thus obtain a third column whose numbers are 'one', 'three', 'three', and 'one'. The table al-Karaji constructed looks like the Pascal triangle on its side.Other results obtained by al-Karaji include summing the first n natural numbers, the squares of the first n natural numbers and the cubes of these numbers. He proved that the sum of the first n natural numbers was n(1/2 + n/2). He also gave (in Rashed and Ahmad's translation, see for example [5]):- The sum of the squares of the numbers that follow one another in natural order from one is equal to the sum of these numbers and the product of each of them by its predecessor. In modern notation this result is i2 = i + i(i-1). Al-Karaji also considered sums of the cubes of the first n natural numbers writing (in Rashed and Ahmad's translation, see for example [5]):- If we want to add the cubes of the numbers that follow one another in their natural order we multiply their sum by itself. In modern notation i3 = ( i)2. Al-Karaji showed that (1 + 2 + 3 + ... + 10)2 was equal to 13 + 23 + 33 + ... + 103. He did this by first showing that (1 + 2 + 3 + ... + 10)2 = (1 + 2 + 3 + ... + 9)2 + 103. He could now use the same rule on (1 + 2 + 3 + ... + 9)2, then on (1 + 2 + 3 + ... + 8)2 etc. to get ( 1 + 2 + ... + 10)2= (1 + 2 + 3 + ... + 8)2 + 93 + 103= (1 + 2 + 3 + ... + 7)2 + 83 + 93 + 103= . . .= 13 + 23 + 33 + ... + 103. Finally we should mention the influence of Diophantus on al-Karaji. The first five books of Diophantus's Arithmetica had been translated into Arabic by ibn Liqa around 870 and these were studied by al-Karaji. Woepcke in his introduction to Al-Fakhri ([7] or [8]) writes that he found:- ... more than a third of the problems of the first book of Diophantus, the problems of the second book starting with the eighth, and virtually all the problems of the third book were included by al-Karaji in his collection. Al-Karaji also invented many new problem of his own but even those of Diophantus were certainly not just taken without further development. He always tried to generalise Diophantus's results and to find methods which were more generally applicable. It was not only to algebra that al-Karaji contributed. The paper [9] discusses some of his geometrical work. This occurs in a chapter entitled On measurement and balances for measuring of buildings and structures. al-Karaji defines points, lines, surfaces, solids and angles. He also gives rules for measuring both plane and solid figures, often using arches as examples. He also gives methods of weighing different substances.Article by: J J O'Connor and E F Robertson July 1999 MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Karaji.html]