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In trigonometry his major contributions are to be found in Ketāb maqālīd ʿelm al-hayʾa (compendium on astronomy), in which he concentrated mainly on the applications of spherical trigonometry in astronomy and provided a detailed classification of spherical triangles and their solutions in Ketāb fī efrād al-maqāl fī amr al-ẓelāl (exhaustive treatise on shadows), in which he developed the familiar trigonometric definitions further and applied them to such religious practices as determining times of prayer and finding the direction of Mecca and in the third book of the Qānūn, in which he propounded trigonometric theorems equivalent to those related to the sums and differences of angles. In determining the mobility of the solar apogee, Bīrūnī followed his Muslim predecessors in departing from the traditional Greek astronomy of Ptolemy, but by means of more refined observational techniques he was able to go farther and to discover that the apogee has a motion of its own, distinct from the motion of precession.
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8) he described the variation in the motion of the sun with respect to the earthly observer in mathematical language that modern historians of science have construed as among the earliest references to mathematical functional relationships (cf. 3), in the course of a discussion devoted to the trigonometric functions used in astronomy, he defined the irrational number pi as the result of division of two other numbers (the circumference of a circle and the diameter), whereas his predecessors, including the Greek authors, had defined it as a geometric ratio. It thus differs from the works of most of Bīrūnī’s predecessors and contemporaries who were concerned only with constructing astronomical tables ( zīj) suitable for computation of planetary positions, usually without any discussion of the derivation of the parameters upon which the tables were based.Īlthough Bīrūnī did not write texts on algebra or geometry and his arithmetical works have not survived, he did introduce new mathematical concepts.
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Like the Almagest, the Qānūn contains theoretical derivations of astronomical parameters, as well as tabular functions to facilitate the computation of planetary positions. Most of Bīrūnī’s original theoretical concepts are to be found in this work. Bīrūnī’s major contribution to astronomy is al-Qānūn al-masʿūdī fi’l-hayʾa wa’l-nojūm (Masʿudic canon of astronomy), covering the same ground as Ptolemy’s Almagest but introducing new material. The following assessment of Bīrūnī’s contributions is based on his work in applied mathematics and on the theoretical portions of his astronomical works. Similarly, although his main concern in astronomy was for computations, he also devoted attention to theoretical problems. The mathematical portions of his works were invariably devoted to applied, rather than theoretical, mathematics nevertheless, in the process of solving problems, Bīrūnī did sometimes indulge in theoretical discussions. Ninety-five of 146 books known to have been written by Bīrūnī, about 65 percent, were devoted to astronomy, mathematics, and related subjects like mathematical geography (Kennedy, p.