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The first operation to be considered is numeration, which is defined in a rather modern vein as the representation of numbers by symbols. To fully appreciate the task the Treviso's author is undertaking in this section, one should understand the level of acceptance for the "Hindu-Arabic" numeral system that existed in Europe at this time (Al-Daffa, 1977).. The new numerals had been known in Europe from about 1000 A.D. yet they had not been universally accepted for use. Computing and the techniques of arithmetic still centered around the manipulation of counters and recording one's results with Roman numerals. There was a certain social status and prestige associated with the use of a counting table.
Hindu mathematics presents interesting features of notation. Valuable information on this development is revealed by the BakhshÄlÄ« Manuscript. First, Hindu Arabic numeral system was mentioned in the 9th century. It is classified as a positional decimal numeral system consisted of symbols (Smith and Karpinski 1911). It has been generally believed that the so-called Arabic numerals, from which arise those in use by us today, were derived by the Muslim peoples from India, and that the Hindus invented (1) the principle of position or place value of the decimal point and (2) the nine digits and zero (or dot). In the astrological treatise written by Ch'-t'an Hsi-ta, who flourished under the T'ang Dynasty in the early eighth century A.D., the so-called Hindu decimal notation and rules are implied, so that they were introduced, or re-introduced into China, at that time or possibly earlier (Datta and Singh 1998).
Whereas Hindu astronomy made improvement through Greek influence, mathematics in India, as Professor Sarton has stated, had no need to wait for Hellenism: we are, therefore, at present disinclined to refuse legitimate claims for Hindu originality in respect of the nine numerals and decimal system (Al-Daffa, 1977). "The basic idea of the system is the primacy of grouping (and of the rhythm of the symbols in their regular sequence) in "packets" of tens, hundreds (tens of tens), thousands (tens of tens of tens), and so on" (Ifrah et al. 2000, p. 25).
In the Bakhshali Manuscript researchers find a small sign used to represent negative quantity: it is a cross, like the present 'plus' sign, but placed to the right of the quantity to which it refers. Zero is represented by a dot. The dot is also used to indicate an unknown quantity. There is an absence from the Bakshali Manuscript of symbols of operation, even the negative sign already noted not being used as such. In Bhaskara's Bija-Ganita, however, the dot is used as the negative sign of operation (Datta and Singh 1998). Operation is indicated in the BakhshÄlÄ« Manuscript by an ad hoc term, or by relative position, In general, Hindu mathematicians used the terms "ya" (as many as") for the first unknown quantity, now usually denoted by x; for the second unknown, say y for the constant quantity in an expression; "v or va" for a square and the initial letters of the words representing various colours for other unknown quantities (Al-Daffa, 1977).
During his khalifate and later, there flourished Al-Kindi. Al-Kindi, like many eminent scholars of the Middle Ages, was an encyclopedialist, and wrote numerous works on many subjects. He translated extesively from the Greek, and his treatise on geometrical and physiological optics (known in the Latin form as De Aspectibus) was based on the optical works of Euclid, Heron, and Ptolemy. He was interested in large-scale natural phenomena, studying particularly the tides, and also the rainbow ion accordance with the principles of optical reflection. Further, his scientific studies embraced the Hindu numerals and a musical notation relating to pitch (Smith and Karpinski 1911). He was sufficiently far-sighted to regard much alchemy as spurious and non-scientific Eminent among writers on mechanical and mathematical subjects were the three sons of MÅ«sÄ ibn ShÄkir, the BanÅ« MÅ«sÄ, who engaged also the great translators H£unain ibn Ish£aq and Thabit ibn Qurra. Among the various writings attributed to the BanÅ« MÅ«sÄ is the Book of the Balance, a treatise dealing with weighing. They also knew the construction of an ellipse by means of a string connecting the foci (Datta and Singh 1998).
There also flourished under Al-Ma'mūn the great mathematician Muhammad ibn Misa al-Khuwarizmi, who was born at Khuwarizm (Khiva) and who was the most influential of mediaeval mathematicians. He fused Hindu and Greek mathematical knowledge, and from his work derive the terms algorism and algebra. His arithmetic introduced the Hindu numeral system to the Arabs and the West; his Algebra gave solutions of linear and quadratic equations, and also a neat geometrical illustration of the solution of the quadratic, x2 + 10x = 39, by 'the completion of the square. (Al-Daffa, 1977).
Mathematicians will know of a further solution, which is x = - 13, but the Arabic scholars dealt only with the positive answer. Unlike the Chinese and Hindus, they tended to ignore negative quantities. Mathematical tables, compiled by AlKhuwarizmÄ« and revised by Al-Majr, contained values of the sine and tangent of angles, and were introduced to Europe in the Latin translation of Adelard of Bath in 1126. Al-KhuwÄrizmÄ« also improved the Geography of Ptolemy.
Albiruni gave the best mediaeval account of the Hindu numerals and the principle of position, and investigated certain mathematical problems which are insoluble by the use of ruler and compasses alone--e.g. the trisection of an angle--and which involve the intersection of conics. This investigation of problems by the intersection of conics, including as it does the solution of cubic equations, was a special feature of mediaeval Arabic mathematics, and several leading mathematicians such as Al-MI+0101hani (Datta and Singh 1998). Al-Khazin, Al-Kuhi, Al-Sijzi, AbÅ«'l Jid, and ' Umar KhayyÄm made their contributions to it. Ibn Al-Banma', the Moroccan mathematician and astronomer, who wrote at least fifty works, including a very popular treatise on the methods of calculation, entitled Talkhi, and containing the use of the Hindu numerals in their Western form, a better treatment of fractions, and the summation of the squares and cubes of the natural numbers; Al- and Muhyi al-Din AlMaghribi, who both worked in the Mongol observatory of Hulagi Khan established at Maraghac. 1259, and Nasir al-Din Al-T (1201-74), the famous director of that observatory; and finally (Boyer, 1991).
While this method is suggested for a two-addend problem, no consideration is given for dealing with a problem with a larger number of addends. The method of casting out nines is very old in the history of arithmetic. Avicenna ( 978-1036) speaks of it as "the method of the Hindus." Transmitted from India by Arab merchants, it became popular in medieval Europe even before its "fellow traveler," the Hindu-Arabic numeral system (Ifrah et al 2000). The acceptance of the use of such a method to assert the correctness of computation in both Eastern and Western cultures was due to the peculiarities of the computational processes used. In the East, calculations made on a dust board or sand table were erased in the course of reaching a solution; similarly, in the West, abacus or counting-table procedures eliminated primary entries in advancing towards an answer. The casting out of nines provided a check on one's work, depending only on a knowledge of the final answer and the initial problem--knowing the intermediate result was not necessary (Datta and Singh 1998).
Since the use of algorithms associated with the Hindu-Arabic numerals, if popularized, could be easily learned and performed without elaborate equipment, its knowledge presented a definite threat to the well-being and livelihood of established computers; therefore, it was resisted. Zero is referred to merely as a symbol to be used in conjunction with digits to express a number (Ifrah et al 2000). By itself, it was literally said to mean nothing, i.e., nulla, nulle, rein, but written to the right of a digit it became a placeholder. Brahmagupta also knew the rules for arithmetical operations involving zero, appreciated the use of negative quantities and of negative terms in algebra (which the Muslim algebraists later ignored), studied quadratic equations, and had considerable success in the solution of indeterminate equations of the second degree. He further stated a rule giving the radius r of the circle circumscribing a triangle ABC (of sides a, b, c) which we now express in the form of its sides in whole numbers, and with the area and circumference of the ellipse; but in the latter investigation his results were inaccurate (Datta and Singh 1998). He also dealt with geometrical progressions. "From this time on, until the decimal system finally adopted the first nine characters and replaced the rest of the Br[=a]hm[=i] notation by adding the zero, the progress of these forms is well marked" (Smith and Karpinski 1911, p. n.a). The Hindus, however, went far beyond the work of Diophantus. Bīja-Ganita contains sections on notation or algorithm; cipher or zero and its use; unknown quantities; surds; the pulverizer; simple and quadratic equations, including a general rule for the solution of the latter which went beyond that of Śrīdhara; solutions of several indeterminate equations of the second degree; and solutions of certain equations of the third and fourth degree (Boyer, 1991).
In sum, the development of the Hindu-Arabic counting system was influenced by mathematical knowledge and development of general sciences. It was not developed as an independent system but was influenced by Hindu and Arabic traditions and knowledge. The "ten place value" notation became a core of modern mathematic and computer science, based on simple symbols and signs.
BIBLIOGRAPHY
Al-Daffa, A. A. The Muslim Contribution to Mathematics. Atlantic Highlands, NJ: Humanities Press.
Boyer, C. B. 1991, A History of Mathematics. New York: Wiley. Revised by Uta C. Merzbach, new ed..
Datta, B., Singh, A. N. 1998, A History of Hindu Mathematics. Bombay: Asia Publishing.
Ifrah, G., et al. 2000, The Universal History of Numbers: From Prehistory to the Invention of the Computer. John Wiley & Sons.
Smith D. E. and Karpinski. L. C. The Hindu-Arabic Numerals. New York, 1911.
HISTORY OF COUNTING SYSTEMS
The first operation to be considered is numeration, which is defined in a rather modern vein as the representation of numbers by symbols. To fully appreciate the task the Treviso's author is undertaking in this section, one should understand the level of acceptance for the "Hindu-Arabic" numeral system that existed in Europe at this time (Al-Daffa, 1977).. The new numerals had been known in Europe from about 1000 A.D. yet they had not been universally accepted for use. Computing and the techniques of arithmetic still centered around the manipulation of counters and recording one's results with Roman numerals. There was a certain social status and prestige associated with the use of a counting table.
Hindu mathematics presents interesting features of notation. Valuable information on this development is revealed by the BakhshÄlÄ« Manuscript. First, Hindu Arabic numeral system was mentioned in the 9th century. It is classified as a positional decimal numeral system consisted of symbols (Smith and Karpinski 1911). It has been generally believed that the so-called Arabic numerals, from which arise those in use by us today, were derived by the Muslim peoples from India, and that the Hindus invented (1) the principle of position or place value of the decimal point and (2) the nine digits and zero (or dot). In the astrological treatise written by Ch'-t'an Hsi-ta, who flourished under the T'ang Dynasty in the early eighth century A.D., the so-called Hindu decimal notation and rules are implied, so that they were introduced, or re-introduced into China, at that time or possibly earlier (Datta and Singh 1998).Whereas Hindu astronomy made improvement through Greek influence, mathematics in India, as Professor Sarton has stated, had no need to wait for Hellenism: we are, therefore, at present disinclined to refuse legitimate claims for Hindu originality in respect of the nine numerals and decimal system (Al-Daffa, 1977). "The basic idea of the system is the primacy of grouping (and of the rhythm of the symbols in their regular sequence) in "packets" of tens, hundreds (tens of tens), thousands (tens of tens of tens), and so on" (Ifrah et al. 2000, p. 25).
In the Bakhshali Manuscript researchers find a small sign used to represent negative quantity: it is a cross, like the present 'plus' sign, but placed to the right of the quantity to which it refers. Zero is represented by a dot. The dot is also used to indicate an unknown quantity. There is an absence from the Bakshali Manuscript of symbols of operation, even the negative sign already noted not being used as such. In Bhaskara's Bija-Ganita, however, the dot is used as the negative sign of operation (Datta and Singh 1998). Operation is indicated in the BakhshÄlÄ« Manuscript by an ad hoc term, or by relative position, In general, Hindu mathematicians used the terms "ya" (as many as") for the first unknown quantity, now usually denoted by x; for the second unknown, say y for the constant quantity in an expression; "v or va" for a square and the initial letters of the words representing various colours for other unknown quantities (Al-Daffa, 1977).
During his khalifate and later, there flourished Al-Kindi. Al-Kindi, like many eminent scholars of the Middle Ages, was an encyclopedialist, and wrote numerous works on many subjects. He translated extesively from the Greek, and his treatise on geometrical and physiological optics (known in the Latin form as De Aspectibus) was based on the optical works of Euclid, Heron, and Ptolemy. He was interested in large-scale natural phenomena, studying particularly the tides, and also the rainbow ion accordance with the principles of optical reflection. Further, his scientific studies embraced the Hindu numerals and a musical notation relating to pitch (Smith and Karpinski 1911). He was sufficiently far-sighted to regard much alchemy as spurious and non-scientific Eminent among writers on mechanical and mathematical subjects were the three sons of MÅ«sÄ ibn ShÄkir, the BanÅ« MÅ«sÄ, who engaged also the great translators H£unain ibn Ish£aq and Thabit ibn Qurra. Among the various writings attributed to the BanÅ« MÅ«sÄ is the Book of the Balance, a treatise dealing with weighing. They also knew the construction of an ellipse by means of a string connecting the foci (Datta and Singh 1998).
There also flourished under Al-Ma'mūn the great mathematician Muhammad ibn Misa al-Khuwarizmi, who was born at Khuwarizm (Khiva) and who was the most influential of mediaeval mathematicians. He fused Hindu and Greek mathematical knowledge, and from his work derive the terms algorism and algebra. His arithmetic introduced the Hindu numeral system to the Arabs and the West; his Algebra gave solutions of linear and quadratic equations, and also a neat geometrical illustration of the solution of the quadratic, x2 + 10x = 39, by 'the completion of the square. (Al-Daffa, 1977).
Mathematicians will know of a further solution, which is x = - 13, but the Arabic scholars dealt only with the positive answer. Unlike the Chinese and Hindus, they tended to ignore negative quantities. Mathematical tables, compiled by AlKhuwarizmÄ« and revised by Al-Majr, contained values of the sine and tangent of angles, and were introduced to Europe in the Latin translation of Adelard of Bath in 1126. Al-KhuwÄrizmÄ« also improved the Geography of Ptolemy.
Albiruni gave the best mediaeval account of the Hindu numerals and the principle of position, and investigated certain mathematical problems which are insoluble by the use of ruler and compasses alone--e.g. the trisection of an angle--and which involve the intersection of conics. This investigation of problems by the intersection of conics, including as it does the solution of cubic equations, was a special feature of mediaeval Arabic mathematics, and several leading mathematicians such as Al-MI+0101hani (Datta and Singh 1998). Al-Khazin, Al-Kuhi, Al-Sijzi, AbÅ«'l Jid, and ' Umar KhayyÄm made their contributions to it. Ibn Al-Banma', the Moroccan mathematician and astronomer, who wrote at least fifty works, including a very popular treatise on the methods of calculation, entitled Talkhi, and containing the use of the Hindu numerals in their Western form, a better treatment of fractions, and the summation of the squares and cubes of the natural numbers; Al- and Muhyi al-Din AlMaghribi, who both worked in the Mongol observatory of Hulagi Khan established at Maraghac. 1259, and Nasir al-Din Al-T (1201-74), the famous director of that observatory; and finally (Boyer, 1991).
While this method is suggested for a two-addend problem, no consideration is given for dealing with a problem with a larger number of addends. The method of casting out nines is very old in the history of arithmetic. Avicenna ( 978-1036) speaks of it as "the method of the Hindus." Transmitted from India by Arab merchants, it became popular in medieval Europe even before its "fellow traveler," the Hindu-Arabic numeral system (Ifrah et al 2000). The acceptance of the use of such a method to assert the correctness of computation in both Eastern and Western cultures was due to the peculiarities of the computational processes used. In the East, calculations made on a dust board or sand table were erased in the course of reaching a solution; similarly, in the West, abacus or counting-table procedures eliminated primary entries in advancing towards an answer. The casting out of nines provided a check on one's work, depending only on a knowledge of the final answer and the initial problem--knowing the intermediate result was not necessary (Datta and Singh 1998).
Since the use of algorithms associated with the Hindu-Arabic numerals, if popularized, could be easily learned and performed without elaborate equipment, its knowledge presented a definite threat to the well-being and livelihood of established computers; therefore, it was resisted. Zero is referred to merely as a symbol to be used in conjunction with digits to express a number (Ifrah et al 2000). By itself, it was literally said to mean nothing, i.e., nulla, nulle, rein, but written to the right of a digit it became a placeholder. Brahmagupta also knew the rules for arithmetical operations involving zero, appreciated the use of negative quantities and of negative terms in algebra (which the Muslim algebraists later ignored), studied quadratic equations, and had considerable success in the solution of indeterminate equations of the second degree. He further stated a rule giving the radius r of the circle circumscribing a triangle ABC (of sides a, b, c) which we now express in the form of its sides in whole numbers, and with the area and circumference of the ellipse; but in the latter investigation his results were inaccurate (Datta and Singh 1998). He also dealt with geometrical progressions. "From this time on, until the decimal system finally adopted the first nine characters and replaced the rest of the Br[=a]hm[=i] notation by adding the zero, the progress of these forms is well marked" (Smith and Karpinski 1911, p. n.a). The Hindus, however, went far beyond the work of Diophantus. Bīja-Ganita contains sections on notation or algorithm; cipher or zero and its use; unknown quantities; surds; the pulverizer; simple and quadratic equations, including a general rule for the solution of the latter which went beyond that of Śrīdhara; solutions of several indeterminate equations of the second degree; and solutions of certain equations of the third and fourth degree (Boyer, 1991).
In sum, the development of the Hindu-Arabic counting system was influenced by mathematical knowledge and development of general sciences. It was not developed as an independent system but was influenced by Hindu and Arabic traditions and knowledge. The "ten place value" notation became a core of modern mathematic and computer science, based on simple symbols and signs.
BIBLIOGRAPHY
Al-Daffa, A. A. The Muslim Contribution to Mathematics. Atlantic Highlands, NJ: Humanities Press.
Boyer, C. B. 1991, A History of Mathematics. New York: Wiley. Revised by Uta C. Merzbach, new ed..
Datta, B., Singh, A. N. 1998, A History of Hindu Mathematics. Bombay: Asia Publishing.
Ifrah, G., et al. 2000, The Universal History of Numbers: From Prehistory to the Invention of the Computer. John Wiley & Sons.
Smith D. E. and Karpinski. L. C. The Hindu-Arabic Numerals. New York, 1911.