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Nasir al-Din al-Tusi. Treatise on spherical trigonometry (untitled in Arabic), illustrated Arabic manuscript on paper, copied by Ahmad ibn Ali ibn Abi al-Faraj ibn al-Bawwab after the author's original, probably Western Persia, Maragha, dated A.H.658/A.D.1260
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Description
56 leaves, 21 to 24 lines per page written in neat naskhi script in brown or black ink on buff paper, headings and significant words in red, numerous diagrams in red or/and black, lacking opening leaf with title-page, later brown morocco binding with stamped central medallion and border, with flap, some wear This lot contains 1 item(s).
Catalogue Note
An important manuscript of the Arabic version of the treatise on spherical trigonometry by the celebrated 13th century polymath Nasir al-Din al-Tusi. It was copied from the author's holograph, completed less than a month earlier, and is the earliest known copy of this text.
Nasir al-Din al-Tusi was the leading Muslim philosopher-scientist of the 13th century. Born in Tus in 597/1201, he studied in Baghdad and Mosul and later worked for the Ismaili rulers at Qayin and Alamut and then, after the Mongol conquest in 654/1256, for their Mongol successor Hulegu in Maragha. He died in Baghdad in 672/1274. Al-Tusi was a truly universal scholar and was perhaps the most prolific author of the Islamic world. He is best known to the history of science for his recensions of early Arabic translations of Greek works on astronomy and mathematics, various independent productions on aspects of theoretical and practical astronomy and mathematics (including the present text) and for the Persian astronomical handbook known as the Zij-i Ilkhani.
In 663/1264-5, al-Tusi, already at Maragha for six years, produced his edition of the Spherics of the second century Greek scholar Menelaos, the most important work of Antiquity on spherical trigonometry. Some years before this, presumably at Qayin or Alamut, he had produced an important independent work on spherical trigonometry. The original work was in Persian, albeit with an Arabic title, Kashf al-qina` an asrar al-qatta`. It survives in what is apparently a unique manuscript in Oxford (MS Bodleian 1498, dated 1100/1688-9). In 658/1260, just a year after the construction of the observatory at Maragha was begun, the author prepared an Arabic version of this text - the al-shakl al qatta`.
Spherical trigonometry is the name given to the methods of solving geometrical problems on a sphere which involves arcs and angles of spherical triangles. In Antiquity the problems were thought of in terms of a transversal of a spherical triangle, the transversal being an arc which cuts each of the three sides (one externally to the triangle): the resulting figure is called a complete spherical quadrilateral. Menelaos derived a powerful theorem relating the ratios of the segments of the sides of the spherical triangle produced by the transversal. His theorem underlies all of the spherical trigonometry in Ptolemy's Almagest, the major Greek work on astronomy. Muslim astronomers inherited the Greek tradition of spherical trigonometry and immediately set upon simplifying it, exploiting special cases of Menelaos' Theorem already discussed in his treatise. From the 9th to the 13th century they produced numerous works on the mathematics of the complete spherical quadrilateral, known in Arabic as al-Shakl al-Qatta`.
Al-Tusi's treatise is now recognized as one of the last and the most complete of this series of works by Muslim scholars on spherical trigonometry. The author himself mentions several of his sources, amongst whom are the well-known scholars Abu Fadl al-Nayrizi, Thabit ibn Qurra and Abu al-Rayhan al-Biruni. Other early scholars who wrote on this subject include the Andalusi Ibn Mu`adh, the celebrated Ibn Sina and Husam al-Salar. The achievement of Muslim scholars in spherical trigonometry was to derive the simpler sine and tangent formulae and use them to solve all the problems of spherical astronomy.
Aside from the present manuscript, previously unrecorded, another dozen or so manuscripts of al-Tusi's text survive. Over a century ago, the earliest of these (dated 677/1278) was used for A.P. Carathéodory's publication of the Arabic text with French translation (Carathéodory, Traité du Quadrilatère attribué à Nassiruddin-El-Toussy, Constantinople, 1891).
Following Carathéodory's text the author divides his work as follows:
'This book contains five parts (maqala) each cont
Nasir al-Din al-Tusi was the leading Muslim philosopher-scientist of the 13th century. Born in Tus in 597/1201, he studied in Baghdad and Mosul and later worked for the Ismaili rulers at Qayin and Alamut and then, after the Mongol conquest in 654/1256, for their Mongol successor Hulegu in Maragha. He died in Baghdad in 672/1274. Al-Tusi was a truly universal scholar and was perhaps the most prolific author of the Islamic world. He is best known to the history of science for his recensions of early Arabic translations of Greek works on astronomy and mathematics, various independent productions on aspects of theoretical and practical astronomy and mathematics (including the present text) and for the Persian astronomical handbook known as the Zij-i Ilkhani.
In 663/1264-5, al-Tusi, already at Maragha for six years, produced his edition of the Spherics of the second century Greek scholar Menelaos, the most important work of Antiquity on spherical trigonometry. Some years before this, presumably at Qayin or Alamut, he had produced an important independent work on spherical trigonometry. The original work was in Persian, albeit with an Arabic title, Kashf al-qina` an asrar al-qatta`. It survives in what is apparently a unique manuscript in Oxford (MS Bodleian 1498, dated 1100/1688-9). In 658/1260, just a year after the construction of the observatory at Maragha was begun, the author prepared an Arabic version of this text - the al-shakl al qatta`.
Spherical trigonometry is the name given to the methods of solving geometrical problems on a sphere which involves arcs and angles of spherical triangles. In Antiquity the problems were thought of in terms of a transversal of a spherical triangle, the transversal being an arc which cuts each of the three sides (one externally to the triangle): the resulting figure is called a complete spherical quadrilateral. Menelaos derived a powerful theorem relating the ratios of the segments of the sides of the spherical triangle produced by the transversal. His theorem underlies all of the spherical trigonometry in Ptolemy's Almagest, the major Greek work on astronomy. Muslim astronomers inherited the Greek tradition of spherical trigonometry and immediately set upon simplifying it, exploiting special cases of Menelaos' Theorem already discussed in his treatise. From the 9th to the 13th century they produced numerous works on the mathematics of the complete spherical quadrilateral, known in Arabic as al-Shakl al-Qatta`.
Al-Tusi's treatise is now recognized as one of the last and the most complete of this series of works by Muslim scholars on spherical trigonometry. The author himself mentions several of his sources, amongst whom are the well-known scholars Abu Fadl al-Nayrizi, Thabit ibn Qurra and Abu al-Rayhan al-Biruni. Other early scholars who wrote on this subject include the Andalusi Ibn Mu`adh, the celebrated Ibn Sina and Husam al-Salar. The achievement of Muslim scholars in spherical trigonometry was to derive the simpler sine and tangent formulae and use them to solve all the problems of spherical astronomy.
Aside from the present manuscript, previously unrecorded, another dozen or so manuscripts of al-Tusi's text survive. Over a century ago, the earliest of these (dated 677/1278) was used for A.P. Carathéodory's publication of the Arabic text with French translation (Carathéodory, Traité du Quadrilatère attribué à Nassiruddin-El-Toussy, Constantinople, 1891).
Following Carathéodory's text the author divides his work as follows:
'This book contains five parts (maqala) each cont