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7.3.5 Nasir al-Din’s Translation of Euclid

15 Rabī‘ II 656 (21 April 1258)
British Library

Naṣīr al-Dīn al-Ṭūsī, Taḥrīr kitāb uṣūl al-handasah wa-al-ḥisāb, an Arabic version of Euclid's fundamental introduction to geometry. Dated 15 Rabī‘ II 656 (21 April 1258), The British Library, Oriental , ADD MS 23387, f. 9v, or

A Greek Hellenistic mathematician, Euclid, wrote The Elements, around 280 BCE. This book was the foundation of geometry. Euclid wrote in Greek, a language that most people in the Abbasid Caliphate could not read. The Abbasid caliph Harun al-Rashid (great-grandson of al-Mansur) began an effort to translate important Greek books into Arabic. A scientist named al-Hajjaj translated Euclid’s Elements into Arabic for Harun al-Rashid around 800 CE. Although almost no books survive from this time, there is evidence that many copies were made of this and other books and sold to the public in Baghdad and other cities of the Abbasid Caliphate. In 1258 another translator, Nasir al-Din, did this Arabic text of Euclid’s Elements. How does this object show interaction and exchange?
Avoid having students read the dense text of this page. The diagrams are enough for them to recognize that this is about geometry. Nothing is known about al-Hajjaj except that he translated this work twice, once for Harun al-Rashid (ruled 786 – 809) and again for his son, al-Mamun. Harun al-Rashid was the first caliph to support scholars in a special library and school in Baghdad and have them translate Greek books about math, science, and medicine into Arabic. This is an exchange of ideas — geometry. The object shows interaction, because the ideas were translated into a new language and spread through books.

المتساوي الأضلاع وننصف زاوية جـ بخط د جـ فينتصف الخط و ذلك لأن في مثلثي ا جـ د بالمطلوب . نريد أن ننصف خطاً محدوداً كخط ا ب فلنعمل عليه مثلث ا جـ ب جـ د ضلعي ا جـ جـ د وزاوية ا جـ د مساوية لضلعي ب جـ جـ د و زاوية ب جـ د فإذن قاعدتا ا د د ب متساويتان و ذلك ما أردناه.
The requested. We want to cut a finite line such as line AB so we construct upon it an equilateral triangle ACB and we cut in half the angle C with line DC then the line is cut in half because in the two triangles ACD and BCD, the legs AC CD and the angle ACD are equal to the legs BC and CD and the angle BCD. Therefore, the bases AD DB are equal and that is what we wanted.

نريد أن نخرج من نقطة على خط غير محدود عموداً عليه مثلا من نقطة جـ على خط اب فلنعين نقطة د كيف وقعت ونجعل جـ ه مثل جـ د و نرسم على د ه مثلث د ر ه المتساوي الأضلاع و نصل ر جـ فهو العمود و ذلك لأن أضلاع مثلثي د ر جـ ه ر جـ متساوية كل لنظيره فزاويتا ر جـ د ر جـ ه الحادثتان عن جانبي ر جـ متساويتان فهما قائمتان و ذلك ما أردناه.
We want to draw from a point on an infinite line a vertical line on it for example from point C on line AB, we specify a point D how it was fallen and make CH like CD and draw on DH equilateral triangle DRH and connect RC that is the vertical line and that is because the legs of the two triangles DRC HRC are equal each to its match thus the two angles RCD RCH that occur on the two sides of RC are equal and they are right and that is what we wanted.

أقول فان كان الخط محدوداً من جانب ا و أردنا أن نخرج العمود من غير إخراج الخط و ذلك مما يحتاج إليه اهل العمل كثيرا فلنعين ح و نجعل جـ د مثل ا جـ و نخرج من جـ د عمودي جـ ه د ر بالوجه المتقدم و ننصف زاويتي ا جـ ه جـ د ر بخطي جـ ح د ه ف جـ ه د ه الخارجتين من خط جـ د الراجح على أقل من قائمتين يتلاقيان بحكم المصادره.
I say if the line is limited from the side A and we wanted to set a vertical line without setting the line and that what is needed a lot by working people we should define E and make the CD like AC and we set from CD verticals CH DR as presented and cut in half the two angles ACH CDR with two lines CE DH so CHDH that are set out of the line CD obviously on less than two right angles that are met by postulation.