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Posted by admin in Recent News on November 28th, 2012

Posted by admin in Uncategorized on November 27th, 2012
The Asian continent is where economic power is shifting from Western Europe and North America. China, India, and South Korea dealt with the global financial crisis of 20078 better than many European states. China is already the second largest world economy; India is noted for having pulled a hundred million people out of extreme poverty. These are impressive achievements even as both China and India are showing signs of a slowdown and the latter still has 450 million people below the poverty line.
Interestingly, China achieved phenomenal economic success without giving up the authoritarian grip of the communist party on power; India has taken great strides forward in the din of its rather chaotic democratic politics. The Asians seem to be able to work any economic model with aplomb.
Economic achievements, however, are not matched by success in creating a harmonious political order in Asia. Bandung’s five principles of coexistence (Panch Sheela) have not led to a resolution of contentious issues left behind by the colonial era. India and Pakistan are the prime examples of this failure as they have fought wars and staged huge military confrontations short of war. China has not been able to settle the border with India not only because of differing territorial claims but also because of unstated approaches to Tibet.
Many Asian analysts argue that the worst violence seen by the continent came from armed interventions by outside powers — the French colonial war in IndoChina, taken over later by the United States; Soviet invasion of Afghanistan; Israel’s westernbacked occupation of Palestine and other invasions of Arab lands; Saddam Hussain’s westerninspired invasion of Iran and American military interventions in Afghanistan and Iraq. According to this interpretation of Asian history, IndiaPakistan conflicts were made more intractable by their conflicting alignments with opposing global power blocs during the Cold War.
The major Asian countries are spending staggering amounts of money on the acquisition and production of arms. China’s gross defence outlay is larger than that of India though, according to Jane’s Defence Weekly, India’s aggregate defence procurement spending between 2011 and 2015 would exceed $100 billion (Dh367 billion). Cashstrapped Pakistan still feels obliged to commit an unaffordable percentage of GDP to national security. Very often, defence expenditures are driven not by real threats but the perception of ruling elites that military power is vital to national esteem, sense of identity and, above all, to the creation of spheres of influence.
As emblems of Asia’s militarisation, indigenously manufactured or purchased missiles abound. International commentaries on the Indian missile test of April 19 highlighted the Indian capability to hit Beijing and China’s great economic hub, Shanghai, with conventional or nuclear weapons. It altered the old maxim that large scale warfare between India and China across the Himalayas was impossible.
By testing AgniV, India has joined the exclusive club of powers capable of producing intercontinental ballistic missiles. It also demonstrated its capability for an ambitious space programme. India already possessed Pakistanspecific Prithvi series, mediumrange Agni series and other effective systems. It can launch conventional and nuclear weapons from land, air and sea.
Outclassed in conventional forces, Pakistan has developed a range of nuclearcapable missiles that effectively counterbalance India. Its medium range Hatf series and a cruise missile can reach most targets in India. Pakistan’s latest test of an improved HatfShaheen A1 missile was carried out only six days after the Indian test. It is also building short range missiles like the 60km Nasr, an entry into the tactical weapons field to bolster its deterrence. Hit hard by Iraq’s Scud missiles, Iranian planners took up missile projects with vengeance. While Iran’s nuclear programme has mysteries, it practices high transparency in missile development as a deterrentenhancer.
Iran’s Shahab3 missile and the Sejjil series cover the Gulf region well and can reach targets in Israel. Of particular interest to Iran’s neighbours and other maritime nations is its arsenal of surface to sea missiles. Iran’s Safir rocket has lifted satellites into space and can, theoretically, lead to long range missiles. Just as India and Pakistan have reached a theatre balance, the regional Arab states are investing heavily in purchasing western missile systems for delivery and interception.
Consider that external actors do not necessarily seek early reduction of tensions. Washington wants to defang North Korea and expects a regime change prior to reconciliation between the two Korean states. It wants India to share the strategic burden of limiting Chinese outreach and influence. Its Iran policy is very complex; its arms deals with Arab states constrain Iranian ambitions and weaken the grip of hawkish elements in the Iranian state. Iran is probably still the main prize that Washington wants to win by diplomacy, coercion or even a limited war.
Deterrence stability has served India and Pakistan well. It can also be effective in the Gulf region. The precondition is robust diplomatic engagement between Iran and the GCC states under a strategic overhang. In the final analysis, missiles will end up as promoters of peace or as instruments of huge damage depending upon how human agents negotiate present differences.
—(Gulf News)
Tanvir Ahmad Khan is a former Pakistani foreign secretary and ambassador to several states.
Posted by admin in Uncategorized on November 27th, 2012
Regarding Pakistan, the rationale of South Korean Official Development Assistance (ODA) and cooperation is not only limited to humanitarian responsibility, rising global issues and increased interdependence among states, rather it is more focused on the responsibility as a past recipient of development assistance from the world including Pakistan in times of need in the 1960s.
The average economic growth rate of Pakistan was higher than the average of the world economy during the 1960s. Average annual real GDP growth was 6.8 percent at that time.
During the same period, Pakistan was seen as a model of economic development around the world, and there was much praise for its economic progress.
Karachi (the largest coastal city of Pakistan) was seen as a global economic role model, and there was much praise for the way its economy was progressing.
Many countries sought to emulate Pakistan’s economic planning strategy and one of them was South Korea, which replicated the second “FiveYear Plan.” The World Financial Center in Seoul was designed and modeled after Karachi.
It is said that Dr. Mahbub ul Haq, a renowned Pakistani economist and the originator of the Human Development Index (HDI), gave this plan to South Korea which helped it to progress rapidly.
At present, many countries envy South Korea, which has progressed in all spheres of national development and ranks as the 26th country in the world with a very high HDI (0.937) as compared to Pakistan which stands at 141st with a medium HDI (0.572) as per the Human Development Report (HDR) 2009 issued by the United Nations Development Program (UNDP).
South Korea graduated from the World Bank’s lending list in 1995 and became a member of OECD donor countries the year after. Pakistan is still striving hard to pass from the lending list of donor agencies.
Therefore, the most valuable asset of South Korea is its experience in making the transition from aid recipient to an emerging donor.
Given the contemporary situation of both countries, Pakistan is also keen to replicate this transformative model and the South Korean government has reciprocated by providing opportunities to Pakistan through various programs under its ODA program and through other cooperation.
The Republic of Korea has provided $20.75 million in aid ($15.94 million in grants and $4.81 million in loans) since 1991 to assist in the development of Pakistan.
Both countries have cooperated on various development activities and have many excellent illustrations for this. The construction of the IslamabadLahore (two major cities of Pakistan) motorway (M2) is one of them.
The construction of the M2 has not only vitalized communication between the two important cities, but also contributed to the enhancement of road construction technology for domestic Pakistani engineers and mechanics that later became able to pursue their own projects without outside help.
It is very encouraging that there are growing numbers of Korean enterprises which are interested in seeking investment opportunities in Pakistan. KP Chemical Corporation, a subsidiary of the South Korean conglomerate Lotte, completed its acquisition of a majority shareholding in Pakistan PTA Ltd (PPTA), investing more than $75 million.
In addition to that, the Korean International Cooperation Agency (KOICA) has planned to invite more Pakistani officials to Korea for capacity building. Some 100 officials will be invited in 2010, and the number will be doubled to 200 in 2011 and 2012. Countryfocused programs will be established to cater for the specific needs of Pakistan.
It is envisaged that in the years to come South Korea will continue reciprocating its assistance and cooperation to Pakistan to an extent that the country will set a worldwide example for countries world that have graduated from the recipients list.
Muhammad Nadeem is a management consultant serving in the public sector organization of Pakistan. He stayed in Republic of Korea recently for training at Central Officials Training Institute (COTI) arranged by the Korea International Cooperation Agency (KOICA) for public sector officials of the member states of the South Asian Association for Regional Cooperation (SAARC). He can be reached at [email protected].
The success of South Korea’s economy in the past 50 years has been remarkable. In 1962 South Korea was among the poorer of the world’s nations, with a per capita income less than Zaire, Congo, and Sudan, and in the next three decades South Korea experienced a growth miracle in which real per capita income increased by about 20 times. In contrast, real per capita income in the United States was about seven times larger at the end of the twentieth century than at the beginning. In percentage terms South Korea’s economic growth in a third of a century far outstripped a century’s worth of US economic growth. This remarkable economic growth began roughly at the same time as President Park’s Third Republic which was established by a military coup in 1961.
President Park designed an economy based on exports. He nationalized banks and set export targets, rewarding those businesspeople who exceeded their targets. High performers were rewarded with economic support such as lowinterest loans and import licenses that would boost their profits. Imports were tightly controlled, exports were subsidized, but exporters were free to import their inputs, dutyfree. By the time President Park was assassinated in 1979, South Korea had been growing at nearly 10% a year for a decade and a half. South Korea’s success in steel production, ship building, automobiles, and eventually electronics moved the nation from the ranks of the povertystricken to one of the world’s leading industrial economies. All of this occurred as the government maintained its policies of supporting successful exporters through financial and regulatory means, lending support to the argument that South Korea’s rapid economic growth was a product of its industrial policy. The idea that industrial policy works by supporting firms that have the potential to be competitive in international markets must be founded on the idea that the government can identify those firms that have this potential. In South Korea firms that were favored by industrial policy were those that were able to demonstrate their ability to export in world markets. What caused successful firms to be successful, and therefore to be favored by the nation’s industrial policy? The answer is entrepreneurship. Firms that were entrepreneurial, that were able to innovate in their production processes by keeping their costs low, and that were able to innovate in their output by producing what consumers wanted, could be competitive in world markets. The initial advantage was produced by entrepreneurship, not be industrial policy.
South Korea had another initial advantage, which was cheap labor, but not all firms were able to use lower labor costs to their advantage. Successful entrepreneurship was what gained firms an edge, and once that edge was demonstrated through their exports; those firms gained further advantages through industrial policy.
In the short run this strategy appears to pay off, because the firms that are the most innovative are the firms that gain additional advantages, which allows them to grow even more than they could have if they were only able to rely on market forces. But innovative firms already have an advantage, and while their initial growth might have been slower without the aid of industrial policy, not only would innovative firms have continued to prosper and grow, they would retain the incentive to be entrepreneurial, because they would only be able to stay on top by being more entrepreneurial than their rivals, both inside and outside South Korea. If they are subsidized by industrial policy, this gives them some slack, and they can remain competitive even if they no longer remain on the cutting edge, because of the advantages they get through government policy.
Industrial policy has two disadvantages. First, it takes away some of the incentive to be entrepreneurial within those firms that industrial policy favors, and second, it makes it more difficult for new firms that may be even more innovative to compete against established firms that have an advantage on an unlevel playing field. For both these reasons, industrial policy has the longrun effect of slowing innovation, and works against the very economic growth it tries to produce. Entrepreneurship and innovation brings with it risktaking. To stay on top of an everchanging world market, firms cannot be content to rest on their past achievements, but must innovate. Sometimes those risks pay off; sometimes they do not. Industrial policy gives firms governmentgranted advantages, which means that firms do not have to take on as much risk, or be as innovative, as whey would without the government support.
In the short run the policy appears to be working because the subsidies go to the firms that have already demonstrated their entrepreneurial nature. In the long run it distorts incentives and will reduce growth. But one key point to see, if one pictures industrial policy as the government’s picking winners and helping them to succeed, is that the government cannot spot those winners until the favored firms have already distinguished themselves through their entrepreneurial actions. It is entrepreneurship, not industrial policy, that provides the initial edge. Once firms find themselves in favored positions industrial policy works against entrepreneurship and continued progress. Firms that were in the cutting edge in the past will not necessarily be the future innovators.
Was South Korea’s Success Produced By Industrial Policy or Entrepreneurship?
Posted on February 1, 2011 by Wyclife Kipruto
The success of South Korea’s economy in the past 50 years has been remarkable. In 1962 South Korea was among the poorer of the world’s nations, with a per capita income less than Zaire, Congo, and Sudan, and in the next three decades South Korea experienced a growth miracle in which real per capita income increased by about 20 times. In contrast, real per capita income in the United States was about seven times larger at the end of the twentieth century than at the beginning. In percentage terms South Korea’s economic growth in a third of a century far outstripped a century’s worth of US economic growth. This remarkable economic growth began roughly at the same time as President Park’s Third Republic which was established by a military coup in 1961.
President Park designed an economy based on exports. He nationalized banks and set export targets, rewarding those businesspeople who exceeded their targets. High performers were rewarded with economic support such as lowinterest loans and import licenses that would boost their profits. Imports were tightly controlled, exports were subsidized, but exporters were free to import their inputs, dutyfree. By the time President Park was assassinated in 1979, South Korea had been growing at nearly 10% a year for a decade and a half. South Korea’s success in steel production, ship building, automobiles, and eventually electronics moved the nation from the ranks of the povertystricken to one of the world’s leading industrial economies. All of this occurred as the government maintained its policies of supporting successful exporters through financial and regulatory means, lending support to the argument that South Korea’s rapid economic growth was a product of its industrial policy. The idea that industrial policy works by supporting firms that have the potential to be competitive in international markets must be founded on the idea that the government can identify those firms that have this potential. In South Korea firms that were favored by industrial policy were those that were able to demonstrate their ability to export in world markets. What caused successful firms to be successful, and therefore to be favored by the nation’s industrial policy? The answer is entrepreneurship. Firms that were entrepreneurial, that were able to innovate in their production processes by keeping their costs low, and that were able to innovate in their output by producing what consumers wanted, could be competitive in world markets. The initial advantage was produced by entrepreneurship, not be industrial policy.
South Korea had another initial advantage, which was cheap labor, but not all firms were able to use lower labor costs to their advantage. Successful entrepreneurship was what gained firms an edge, and once that edge was demonstrated through their exports; those firms gained further advantages through industrial policy.
In the short run this strategy appears to pay off, because the firms that are the most innovative are the firms that gain additional advantages, which allows them to grow even more than they could have if they were only able to rely on market forces. But innovative firms already have an advantage, and while their initial growth might have been slower without the aid of industrial policy, not only would innovative firms have continued to prosper and grow, they would retain the incentive to be entrepreneurial, because they would only be able to stay on top by being more entrepreneurial than their rivals, both inside and outside South Korea. If they are subsidized by industrial policy, this gives them some slack, and they can remain competitive even if they no longer remain on the cutting edge, because of the advantages they get through government policy.
Industrial policy has two disadvantages. First, it takes away some of the incentive to be entrepreneurial within those firms that industrial policy favors, and second, it makes it more difficult for new firms that may be even more innovative to compete against established firms that have an advantage on an uneven playing field. For both these reasons, industrial policy has the longrun effect of slowing innovation, and works against the very economic growth it tries to produce. Entrepreneurship and innovation brings with it risktaking. To stay on top of an everchanging world market, firms cannot be content to rest on their past achievements, but must innovate. Sometimes those risks pay off; sometimes they do not. Industrial policy gives firms governmentgranted advantages, which means that firms do not have to take on as much risk, or be as innovative, as whey would without the government support.
In the short run the policy appears to be working because the subsidies go to the firms that have already demonstrated their entrepreneurial nature. In the long run it distorts incentives and will reduce growth. But one key point to see, if one pictures industrial policy as the government’s picking winners and helping them to succeed, is that the government cannot spot those winners until the favored firms have already distinguished themselves through their entrepreneurial actions. It is entrepreneurship, not industrial policy, that provides the initial edge. Once firms find themselves in favored positions industrial policy works against entrepreneurship and continued progress. Firms that were in the cutting edge in the past will not necessarily be the future innovators.
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Recent research paints a new picture of the debt that we owe to Arabic/Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects the mathematics studied today is far closer in style to that of the Arabic/Islamic contribution than to that of the Greeks.
There is a widely held view that, after a brilliant period for mathematics when the Greeks laid the foundations for modern mathematics, there was a period of stagnation before the Europeans took over where the Greeks left off at the beginning of the sixteenth century. The common perception of the period of 1000 years or so between the ancient Greeks and the European Renaissance is that little happened in the world of mathematics except that some Arabic translations of Greek texts were made which preserved the Greek learning so that it was available to the Europeans at the beginning of the sixteenth century.
That such views should be generally held is of no surprise. Many leading historians of mathematics have contributed to the perception by either omitting any mention of Arabic/Islamic mathematics in the historical development of the subject or with statements such as that made by Duhem in [3]:
… Arabic science only reproduced the teachings received from Greek science.
Before we proceed it is worth trying to define the period that this article covers and give an overall description to cover the mathematicians who contributed. The period we cover is easy to describe: it stretches from the end of the eighth century to about the middle of the fifteenth century. Giving a description to cover the mathematicians who contributed, however, is much harder. The works [6] and [17] are on “Islamic mathematics”, similar to [1] which uses the title the “Muslim contribution to mathematics”. Other authors try the description “Arabic mathematics”, see for example [10] and [11]. However, certainly not all the mathematicians we wish to include were Muslims; some were Jews, some Christians, some of other faiths. Nor were all these mathematicians Arabs, but for convenience we will call our topic “Arab mathematics”.
The regions from which the “Arab mathematicians” came was centred on Iran/Iraq but varied with military conquest during the period. At its greatest extent it stretched to the west through Turkey and North Africa to include most of Spain, and to the east as far as the borders of China.
The background to the mathematical developments which began in Baghdad around 800 is not well understood. Certainly there was an important influence which came from the Hindu mathematicians whose earlier development of the decimal system and numerals was important. There began a remarkable period of mathematical progress with alKhwarizmi‘s work and the translations of Greek texts.
This period begins under the Caliph Harun alRashid, the fifth Caliph of the Abbasid dynasty, whose reign began in 786. He encouraged scholarship and the first translations of Greek texts into Arabic, such as Euclid‘s Elements by alHajjaj, were made during alRashid’s reign. The next Caliph, alMa’mun, encouraged learning even more strongly than his father alRashid, and he set up the House of Wisdom in Baghdad which became the centre for both the work of translating and of of research. AlKindi (born 801) and the three Banu Musa brothers worked there, as did the famous translator Hunayn ibn Ishaq.
We should emphasise that the translations into Arabic at this time were made by scientists and mathematicians such as those named above, not by language experts ignorant of mathematics, and the need for the translations was stimulated by the most advanced research of the time. It is important to realise that the translating was not done for its own sake, but was done as part of the current research effort. The most important Greek mathematical texts which were translated are listed in [17]:
Of Euclid‘s works, the Elements, the Data, the Optics, the Phaenomena, and On Divisions were translated. Of Archimedes‘ works only two – Sphere and Cylinder and Measurement of the Circle – are known to have been translated, but these were sufficient to stimulate independent researches from the 9^{th} to the 15^{th} century. On the other hand, virtually all of Apollonius‘s works were translated, and of Diophantus and Menelaus one book each, the Arithmetica and the Sphaerica, respectively, were translated into Arabic. Finally, the translation of Ptolemy‘s Almagest furnished important astronomical material.
The more minor Greek mathematical texts which were translated are also given in [17]:
… Diocles‘ treatise on mirrors, Theodosius‘s Spherics, Pappus‘s work on mechanics, Ptolemy‘s Planisphaerium, and Hypsicles‘ treatises on regular polyhedra (the socalled Books XIV and XV of Euclid‘s Elements) …
Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of alKhwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry.
Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as “algebraic objects”. It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before. As Rashed writes in [11] (see also [10]):
AlKhwarizmi‘s successors undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.
Let us follow the development of algebra for a moment and look at alKhwarizmi‘s successors. About forty years after alKhwarizmi is the work of alMahani (born 820), who conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Abu Kamil (born 850) forms an important link in the development of algebra between alKhwarizmi and alKaraji. Despite not using symbols, but writing powers of x in words, he had begun to understand what we would write in symbols as x^{n}.x^{m} = x^{m+n}. Let us remark that symbols did not appear in Arabic mathematics until much later. Ibn alBanna and alQalasadi used symbols in the 15^{th} century and, although we do not know exactly when their use began, we know that symbols were used at least a century before this.
AlKaraji (born 953) is seen by many as the first person to completely free algebra from geometrical operations and to replace them with the arithmetical type of operations which are at the core of algebra today. He was first to define the monomials x, x^{2}, x^{3}, … and 1/x, 1/x^{2}, 1/x^{3}, … and to give rules for products of any two of these. He started a school of algebra which flourished for several hundreds of years. AlSamawal, nearly 200 years later, was an important member of alKaraji‘s school. AlSamawal (born 1130) was the first to give the new topic of algebra a precise description when he wrote that it was concerned:
… with operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.
Omar Khayyam (born 1048) gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. Khayyam also wrote that he hoped to give a full description of the algebraic 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.
Sharaf alDin alTusi (born 1135), although almost exactly the same age as alSamawal, does not follow the general development that came through alKaraji‘s school of algebra but rather follows Khayyam‘s application of algebra to geometry. He wrote a treatise on cubic equations, which [11]:
… represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.
Let us give other examples of the development of Arabic mathematics. Returning to the House of Wisdom in Baghdad in the 9^{th} century, one mathematician who was educated there by the Banu Musa brothers was Thabit ibn Qurra(born 836). He made many contributions to mathematics, but let us consider for the moment consider his contributions to number theory. He discovered a beautiful theorem which allowed pairs of amicable numbers to be found, that is two numbers such that each is the sum of the proper divisors of the other. AlBaghdadi (born 980) looked at a slight variant of Thabit ibn Qurra‘s theorem, while alHaytham (born 965) seems to have been the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors) as those of the form 2^{k1}(2^{k} – 1) where 2^{k} – 1 is prime.
AlHaytham, is also the first person that we know to state Wilson’s theorem, namely that if p is prime then 1+(p1)! is divisible by p. It is unclear whether he knew how to prove this result. It is called Wilson’s theorem because of a comment made by Waring in 1770 that John Wilson had noticed the result. There is no evidence that John Wilson knew how to prove it and most certainly Waring did not. Lagrange gave the first proof in 1771 and it should be noticed that it is more than 750 years after alHaytham before number theory surpasses this achievement of Arabic mathematics.
Continuing the story of amicable numbers, from which we have taken a diversion, it is worth noting that they play a large role in Arabic mathematics. AlFarisi (born 1260) gave a new proof of Thabit ibn Qurra‘s theorem, introducing important new ideas concerning factorisation and combinatorial methods. He also gave the pair of amicable numbers 17296, 18416 which have been attributed to Euler, but we know that these were known earlier than alFarisi, perhaps even by Thabit ibn Qurra himself. Although outside our time range for Arabic mathematics in this article, it is worth noting that in the 17^{th} century the Arabic mathematician Mohammed Baqir Yazdi gave the pair of amicable number 9,363,584 and 9,437,056 still many years before Euler‘s contribution.
Let us turn to the different systems of counting which were in use around the 10^{th} century in Arabic countries. There were three different types of arithmetic used around this period and, by the end of the 10^{th} century, authors such as alBaghdadi were writing texts comparing the three systems.
1. Fingerreckoning arithmetic.
This system derived from counting on the fingers with the numerals written entirely in words; this fingerreckoning arithmetic was the system used by the business community. Mathematicians such as Abu’lWafa (born 940) wrote several treatises using this system. Abu’lWafa himself was an expert in the use of Indian numerals but these:
… did not find application in business circles and among the population of the Eastern Caliphate for a long time.
Hence he wrote his text using fingerreckoning arithmetic since this was the system used by the business community.
2. Sexagesimal system.
The second of the three systems was the sexagesimal system, with numerals denoted by letters of the Arabic alphabet. It came originally from the Babylonians and was most frequently used by the Arabic mathematicians in astronomical work.
3. Indian numeral system.
The third system was the arithmetic of the Indian numerals and fractions with the decimal placevalue system. The numerals used were taken over from India, but there was not a standard set of symbols. Different parts of the Arabic world used slightly different forms of the numerals. At first the Indian methods were used by the Arabs with a dust board. A dust board was needed because the methods required the moving of numbers around in the calculation and rubbing some out as the calculation proceeded. The dust board allowed this to be done in the same sort of way that one can use a blackboard, chalk and a blackboard eraser. However, alUqlidisi (born 920) showed how to modify the methods for pen and paper use. AlBaghdadi also contributed to improvements in the decimal system.
It was this third system of calculating which allowed most of the advances in numerical methods by the Arabs. It allowed the extraction of roots by mathematicians such as Abu’lWafa and Omar Khayyam (born 1048). The discovery of the binomial theorem for integer exponents by alKaraji (born 953) was a major factor in the development of numerical analysis based on the decimal system. AlKashi (born 1380) contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers such as π. His contribution to decimal fractions is so major that for many years he was considered as their inventor. Although not the first to do so, alKashi gave an algorithm for calculating nth roots which is a special case of the methods given many centuries later by Ruffini and Horner.
Although the Arabic mathematicians are most famed for their work on algebra, number theory and number systems, they also made considerable contributions to geometry, trigonometry and mathematical astronomy. Ibrahim ibn Sinan(born 908), who introduced a method of integration more general than that of Archimedes, and alQuhi (born 940) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particular alHaytham, studied optics and investigated the optical properties of mirrors made from conic sections. Omar Khayyam combined the use of trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means.
Astronomy, timekeeping and geography provided other motivations for geometrical and trigonometrical research. For example Ibrahim ibn Sinan and his grandfather Thabit ibn Qurra both studied curves required in the construction of sundials. Abu’lWafa and Abu Nasr Mansur both applied spherical geometry to astronomy and also used formulas involving sin and tan. AlBiruni (born 973) used the sin formula in both astronomy and in the calculation of longitudes and latitudes of many cities. Again both astronomy and geography motivated alBiruni‘s extensive studies of projecting a hemisphere onto the plane.
Thabit ibn Qurra undertook both theoretical and observational work in astronomy. AlBattani (born 850) made accurate observations which allowed him to improve on Ptolemy‘s data for the sun and the moon. Nasir alDin alTusi (born 1201), like many other Arabic mathematicians, based his theoretical astronomy on Ptolemy‘s work but alTusi made the most significant development of Ptolemy‘s model of the planetary system up to the development of the heliocentric model in the time of Copernicus.
Many of the Arabic mathematicians produced tables of trigonometric functions as part of their studies of astronomy. These include Ulugh Beg (born 1393) and alKashi. The construction of astronomical instruments such as the astrolabe was also a speciality of the Arabs. AlMahani used an astrolabe while Ahmed (born 835), alKhazin (born 900), Ibrahim ibn Sinan, alQuhi, Abu Nasr Mansur (born 965), alBiruni, and others, all wrote important treatises on the astrolabe.Sharaf alDin alTusi (born 1201) invented the linear astrolabe. The importance of the Arabic mathematicians in the development of the astrolabe is described in [17]:
The astrolabe, whose mathematical theory is based on the stereographic projection of the sphere, was invented in late antiquity, but its extensive development in Islam made it the pocket watch of the medievals. In its original form, it required a different plate of horizon coordinates for each latitude, but in the 11^{th} century the Spanish Muslim astronomer azZarqallu invented a single plate that worked for all latitudes. Slightly earlier, astronomers in the East had experimented with plane projections of the sphere, and alBiruni invented such a projection that could be used to produce a map of a hemisphere. The culminating masterpiece was the astrolabe of the Syrian Ibn ashShatir (130575), a mathematical tool that could be used to solve all the standard problems of spherical astronomy in five different ways.
References (21 books/articles)
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Article by: J J O’Connor and E F Robertson