This article aims to give an overview of the formation and development of mathematical studies and the work of famous mathematician in the Ottoman State over a 600 year period, from the period preceding the conquest of Constantinople to the early 20th century. Dozens of mathematicians and hundreds of mathematical works flourished and they constitute rich material for ongoing investigation.
Dr. Salim Ayduz*
Table of contents
4.1. Qādīzāda al-Rūmī (d. ca 1440)
4.2. ‘Alā al-Dīn ‘Ali al-Qushjī (ca 1402-1474)
4.3. Khalil al-Husayni (15th century)
4.4. Yusuf Sinan Pasha (d. 1486)
4.5. Hajji Atmaca al-Kātib (d. after 1494)
4.6. Lutfullah al-Toqātī (d. 1494)
4.7. Mīrīm Celebi (d. 1525)
4.8. Eliya Mizrahi (ca 1450-1526)
4.9. Nasūh ‘Alī al-Silāhī al-Matrākī (d. 1564)
4.10. Taqī al-Dīn ibn Ma'rûf (1520 - 1585)
5.1. Khalil Fā'id Efendi
5.2. As'ad Efendi al-Yanyawī (Yanyali Esad Efendi)
5.3. Muhammad Istanbulī
5.4. ‘Abd Al-Rahīm Al-Mar'ashi Efendi (d. 1736)
5.5. Mustafa Sidkī b. Sālih Kethüdā (d. 1183/1769)
5.6. Ibrāhim of Aleppo (d. 1776)
5.7. Sekerzāda Sayyid Fayzullah Sarmad (d. 1787)
5.8. Gelenbevī Ismail Efendi (1730-1790)
5.9. Kalfazāda/Halifezāda Ismail Efendi (d. 1790)
5.10. Huseyin Rifki Tāmānī (d. Madina, 1817)
6.1. Hoca Ishak Efendi
6.2. Ahmad Tawfīk Efendi (1807-1869)
6.3. Hüseyin Tevfik Pasha of Vidin (1832-1893)
6.4. Salih Zeki (1845-1921)
The Ottoman state began as a local principality at the turn of the 14th century. It became the most powerful state over a vast area extending from Central Europe to the Indian Ocean. During the 600 years of its existence, alongside the political events, the development of scientific and cultural activities played a crucial role on the cultural and scientific fronts. As a continuation of previous Muslim-Turkic states, the Ottomans inherited scientific and cultural riches from the Golden Age of Muslim Civilisation, from the 9th century onwards. With this legacy, they improved and established their own schools. Scientific activities emerged and developed from the base of the pre-Ottoman Seljukid period in Anatolian cites by benefiting from the works of scholars, who came from different corners of Islamic lands such as Egypt, Syria, Iran, India and Turkestan. The new Ottoman scholars and intellectuals, brought a new enthusiasm to cultural and scientific life. Beside Istanbul, many new centers flourished throughout the Ottoman lands, particularly in the Balkans and other European territories in places such as Bursa, Edirne, Istanbul, Skopje, and Sarajevo. This article aims to give an overview of the formation and development of mathematical studies and present bio-bibliographies of some famous Ottoman mathematicians over a six hundred years period. **
Figure 1: An Ottoman miniature showing a game of Matrak invented by Nasūh ‘Alī al-Silāhī al-Matrāqī. Source: Topkapi Palace library in Istanbul, MS H 1344.
Most of the scholars during the first two centuries of the Ottomans came from Muslim countries and Turkish municipalities. The first Ottoman school (madrasa, pl. madrasas) was built in Iznik (Nicea)  in 1331 by the second Ottoman ruler Gazi Orhan Beg (c. 1326-1359) just after he conquered the city in 1331. Gazi Orhan Beg established many foundations in order to meet the financial needs of the madrasa. The Iznik madrasa trained the student in religious sciences (al-'ulūm al-diniyya) in their totality, and famous religious scholars such as Dāwūd al-Qaysarī (d. 1350) , Tāj al-Dīn al-Garadī (d. c. 1360) and Ala al-Dīn Aswad (d. 1393) taught in this madrasa. After the conquest of Bursa and Edirne, new schools and other educational buildings such as medical institutes and primary schools opened and scholars started to flock to the Ottoman cities. Scholars from different backgrounds produced very important books on various subjects including mathematics and astronomy. In this study, we will analyze the research and teaching achievements performed by eminent Ottoman mathematicians and study their works.
The turning point in the history of madrasa teaching and the shift from the traditional Nizamiya madrasa system towards a more comprehensive institutional model took place during the time of Muhammad II. There are several reasons for this new orientation. First, it was the personal interest of Sultan himself in the rational sciences and his support of the scholars; second, this new tradition seems to draw directly from the Ilkhanid and Timurid institutions of learning, which included the teaching of the rational sciences. Muhammad II founded the Fatih Complex, which bore his name, between 1463 and 1470. It included eight intermediate madrasas called ‘tatimma‘ and eight other high madrasas called ‘sahn‘ (literally courtyard).
The history of mathematical and astronomical literature during the Ottoman period records that numerous copies of astronomical and mathematical works were produced in the madrasa system. On the other hand, we note that from the 16th until the 19th century, there was an increase in the number produced .
For instance, Qādīzāda's (d. 1432) two works on astronomy and mathematics Sharh al-Mulakhkhas fī al-Hay'a and Tuhfat al-Ra'is fī Sharh Ashkāl al-Ta'sīs were two basic textbooks for students who wished to study these subjects. There are more than three hundred extant copies of the former and approximately two hundred copies of the latter. Among these copies there are a considerable numbers which were copied in the schools of Anatolia and Istanbul .
After the conquest of Istanbul by Muhammad the Conqueror (1451-1481) in 1453, the Sultan himself began to set up a science centre in Istanbul. In the Library of the Palace of the Sultan, we find copies of numerous books about medicine, arithmetic, geometry, astronomy which were published in other countries during his time. During his reign, Muhammad II invited famous scholars to study in Istanbul at his Madrasas. During his reign, new educational institutions, such as the Sahn-i Sāmān Madrasas and the Enderūn Palace School in Istanbul, were established. As a result, some brilliant scholars emerged and made original contributions to science during his reign. The works of ‘Ali al-Qushjī (d. 1474) and Fathullah al-Shirwānī's (d. 1486) two students of Qādīzāda al-Rūmī (d. ca 1440) from Samarkand, made notable mathematical contributions. Muhammad II patronized Muslim and non-Muslim scholars in Istanbul and ordered Greek scholars to translate Ptolemy's Geography into Arabic and to draw a world map. In addition to Muslim scholars from the Muslim world, he also invited artists and scholars from Europe especially Italy. Muhammad II also encouraged the scholars of his time to produce works in their fields.
In the Ottoman school system, mathematics and geometry were studied before the Hadith and the Quran studies. Muhammad b. Abu Bakr al-Marashi stated in his book Tartīb al-'Ulûm (written in 1715) that in the Ottoman schools, the students could learn geometry, cosmology and literature at any time, but arithmetic had to be studied as a compulsory science by all Muslims. It is possible to understand from the autobiographies of Ottoman scholars how the mathematic courses were planned in the curriculum. In the autobiography of Sheik al-Islam Feyzullah Efendi (d. 1703) it was stated that arithmetic, geometry and astronomy courses were taught with the courses of Hikma (wisdom) and Tafsir (explanation of the Quran) . There are some records stating that arithmetic was also taught in religious institutions such as the takkas and zaviyas (Darwish lodges) .
In his book De La Littérature des Turcs, Abbé Toderini (lived in Istanbul between 1781 and 1786) stated that the Turks learnt arithmetic from well-written Turkish -Arabic course books and were as well informed as a European mathematicians. In the geometry section of De La Littérature, Toderini describes geometry instruction in the Ottoman madrasas:
"Geometry falls under the group of Turkish studies. In academies (madrasa), there are professors (mudarris) for teaching it [geometry] to young people. The period between mathematics and rhetoric classes is allocated to this mathematical branch... This science is taught in a special manner. I have been to the Valide Madrasa twice, during which time students had gathered to listen to the geometry class. They used an Arabic translation of Euclid. There are many versions as well as commentaries of this book. Nasīr al-Dīn al-Ţūsī's commentary, which is regarded as the best of these, has already become popular thanks to the Medicis Publishing House. This copy contains a copy of the Turkish license granted by Sultan Murad III (1574-1595) in Istanbul in 1587 . He has granted permission for the sale of this book without any tax or liability within the entire Ottoman territory..." 
Students in the Ottoman schools were allowed to teach arithmetic under the supervision of their teachers after a certain level of education. This means that after theoretical education the students had the opportunity to apply their knowledge. It is observed that in 19th-century mathematics education also became important in high schools. It is known that some of the Ottoman scholars learnt higher mathematics and algebra in high schools.
The Ottoman scholars wrote many textbooks on mathematics and also translated other important ones written in other countries. In arithmetic, they mostly used books written by Muslim mathematicians. For instance, al-Muhammadiyya fī al-Hisāb written by Ali al-Qushjī and Khulāsāt al-Hisāb written by Bahā al-Dīn al-'Āmilī were the widely used course books in arithmetic. Today there are more than forty copies of the Muhammadiyya in the libraries of various countries. We know of sixteen Arabic copies in Turkish libraries and two other ones in Cairo and in Aleppo .
Bahā al-Dīn al-'Āmilī's Khulāsāt al-Hisāb was a course book throughout the Ottoman school system from the 17th century. Kuyucaklizāda Muhammed Ātif (d. 1847) translated the book into Turkish with commentaries as Nihāyat al-Albāb fī Tarjamati Khulāsāt al-Hisāb. This was in 1826 during the reign of Mahmud II following his request to understand the original book easily . For three centuries, this was one of the most common textbooks for students. It was studied as a textbook in the Ottoman State, Persia, India, and Egypt and it was translated into German in 1843 by Nesselman and also into French in 1846 by A. Marre. Some books written in Europe in the 19th century contained quotations from Khulāsāt al-Hisāb. The final editions of the book were in Istanbul in 1879 and in Cairo in 1894. There are more than one hundred copies of the book in Turkish libraries .
The Ottoman scholars started to write arithmetic books from the beginning of the 15th century onwards. Arithmetical texts were translated into Turkish after those of astronomy, but before texts of geometry. The titles of some are Miftāh al-Hisāb (anonymous), Risāla fī ilm-i Hisāb (anonymous) and Miftāh al-Mushkilāt (Muhammed Musa-i Wāfī). The arithmetic books which were prepared by the Muhasipler (account scribes) and diwan katipleri (secretaries of the Council of State) were usually written in Turkish.
The book entitled Majma'-I qawā'id-i ‘ilm-i hisāb written by Hajji Atmaca in 1484 is an example of this. The greatest among the Turkish books of arithmetic which was written in the classical tradition, was Tuhfat al-A'dad li-zawi al-rushd wa al-sadād written by Ali b. Veli Hamza b. al-Jazāirī al-Maghrībī (d. 1614) in 1590 and was presented to Sultan Murād III (d. 1595). This book shows that symbols and notation in algebra were used commonly by Ottoman mathematicians . The other famous arithmetical books are Nuzhat al-Hussab fī ‘ilm al-Hisāb, al-Luma fī al-Hisāb, al-Ma'una fī al-Hisāb al-Hawa'l written by Ibn al-Ha'im (d. 1412) and Talkhīs A'māl al-Hisāb written by the Moroccan mathematician Ibn al-Bannā (d. 1321). Although these authors were not strictly Ottoman citizens, their books were common in the Ottoman lands and were widely read and translated.
The Ottoman scholars also wrote and translated course books about algebra. Some of the book relating to algebra are Al-Yawakit Al-Mufassalāt bi-'l-La'ali al-Nayyirāt fī A'māli Zawāt al-Asmā wa-'l-Munfasilāt written by Jamal Al-Dīn Muhammed b. Ahmad b. Muhammad b. Pīrī al-'Alwānī, also known as Ibn Pīrī (d. 1631) ; Al-Mawāhib al-Saniyya fī Ilm al-Jabr wa-'l-Muqābala and Sharh al-Yasminiya fī al-Jabr wa-'l-Muqābala written by Ibn Al-Jamal (d. 1662); al-Mustahzarat fī Hisāb al-Majhulāt by Kuyucaklizade Muhammed Atif (d. 1847) and Tuhfat al-Hisāb by Ali Bahar Efendi (d. 1805) .
The Ottoman scholars were interested in logarithm because of the preparation of the star tables. They wrote and translated some books about logarithm. In 1780 Sekerzāda Feyzullah Sermed (d. 1787) translated the book entitled Maqsadayn fī Hall Al-Nisbatayn from a Hungarian mathematician. In this book, he defined the logarithm and explained the applications of logarithm in astronomy. The other books written by Ottoman scholars about logarithm are Sharh al-Jadāwil al-Ansāb by Gelenbevī Ismail Efendi in 1787; Logaritma Risālasi by Huseyin Rifki Tāmānī (d. 1817) and Logaritma Risālasi by Muftuzade Osman Saib (d. 1864) .
The Hendesehāne-i Humayun (Royal Mathematical School) was the first institution that was designated for modern military technical education in the Ottoman State. The Hendesehāne, which was called the ‘Ecole des Théories‘ or the ‘Ecoles des Mathématiques‘ in French, was established at the Royal Shipyard on 29 April 1775. In addition to the Ottoman teachers, Baron de Tott and a French expert taught courses. The institution had up to ten students and later assumed the name of the Muhendishāne-i Humāyûn (Royal School of Engineering) .
A great number of French and a few English engineers, teachers and officers came to Istanbul between 1783 and 1788, with the renewed closeness between the Ottomans and the French. However, all the French experts and foremen left Istanbul as the result of the alliance formed between Russia and France when the Ottomans entered into war against Russia between 1787 and 1788 . It was observed that foremen and workers from other European states (some from Sweden) were employed after the French departed. When all the French experts and officers returned to their country between 1787 and 1788, the applied mathematics courses were discontinued and only the theoretical mathematics courses continued, given by Ottoman scholars, such as Gelenbevī Ismāil Efendi (d. 1790) and Palabiyik Muhammad Efendi (d. 1804).
Figure 2, 3 & 4: Sample pages of Taqī al-Dīn ibn Ma'rūf's Hisāb al-munanjjimīn wa-'l-jabr wa-'l-muqābala. Source: Süleymaniye library in Istanbuly, Carullah collection, MS 1454.
4.1. Qādīzāda al-Rūmī (d. ca 1440)
His full name was Salah al-Dīn Musa ibn Muhammad ibn Mahmud Qādīzāda al-Bursāwi al-Rūmī. He was born in Bursa, Turkey, (hence his name al-Rūmī, from the Arabic name al-Rum for the Byzantine and Ottoman States). His grandfather and father were judges/Qādī in Bursa. He received his preliminary education in mathematics and cosmology in the province of Bursa and then went to Samarkand. He became the teacher of Ulugh Beg (d. 1449) in astronomy and later on was appointed the chief instructor at the school of Samarkand and the director of the observatory founded by Ulugh Beg (d. 1449). He died there and was buried by Ulugh Beg in the mausoleum of Shāh-i-zinda (Living King) in Samarkand.
Qādīzāda al-Rūmī made the first important contribution to the development of the Ottoman scientific tradition and literature on mathematics and astronomy. He flourished in Anatolia and settled in Samarkand after compiling his first work. He wrote Sharh Mulakhkhas fī'l-hay'a (Commentary on the ‘Compendium on Astronomy') and Sharh Ashkāl al-Ta'sīs (Commentary on The Fundamental Theorems) in Arabic in the fields of astronomy and mathematics. He simplified the calculation of the sine of a one degree arc in his work Risāla fī Istikhrāj Jaybi Daraja Wāhida (Treatise on the Calculation of the Sine of a One Degree of the Arc).
Qādīzāda's two students ‘Ali al-Qushjī (d. 1474) and Fathullah al-Shirwānī (d. 1486) influenced Ottoman science by disseminating work on mathematics and astronomy. In the introduction to his Tuhfat al-Ra'is Fi Sharh Ashkāl al-Ta'sīs (Gift of the Chief in the Commentary on The Fundamental Theorems), he indicated that the philosophers who ponder about the creation and the secrets of the universe, the jurists (faqihs) who give fatwās in religious matters, the officials who run the affairs of state, and the qādīs who deal with judicial matters should know geometry. Thus, he emphasized the necessity of science to philosophical, religious, and worldly matters. This understanding reflects a general characteristic of Ottoman science.
In addition to the above, Qādīzāda wrote other books on mathematics and astronomy, made significant contribution to the preparation of Ulugh Beg's Zij  and wrote many commentaries on astronomical and geometry books. We present them below.
- Tuhfat al-Ra'is mentioned above : Qādīzāda wrote this commentary on Samarkandī's Ashqāl al-ta'sis which is a summary of the theorems and triangles in Euclid's Usûl al-handasa. It was completed in 1412 and presented to Ulugh Beg. There are approximately two hundred copies of this treatise in libraries .
- Risāla dar bayān-i istikhrāj jayb-i yak daraja (Treatise on Explanation of Determining the Sine of One Degree) by operations based on rules that are based on Arithmetic and geometric by principles of the Method of Ghiyāth al-Dīn al-Kāshī. Although it is a commentary on the treatise of al-Kāshī titled Risāla al-watar wa'l-jayb (Treatise on Chord and Sine), due to the originality of the subject, al- Rūmī is often regarded as the author of the treatise . According to Salih Zeki (d. 1921), it is the most important treatise by him .
- Risāla fī al-misāha (Treatise on surveying), in Persian: In the prologue of the treatise, al-Rūmī explains why he composed this book saying that "some of my friends and tax officials asked me to write a treatise to solve their problems on the measurement (surveying) calculations. Therefore I composed this treatise." The treatise was divided into four chapters (Ruqun) and twelve sections (qāidah) .
4.2. ‘Alā al-Dīn ‘Ali al-Qushjī (ca 1402-1474)
This scholar's full name is Qushci-zāda Abu al-Qāsim ‘Alā al-Dīn Ali b. Muhammad. He was born in Samarkand in the early 15th century. His father was Ulugh Beg's official falcon trainer; he came to be known as "Qushci-zāda" or "Qushjī." He received advanced education from outstanding scholars such as Ulugh Beg, Giyāth al-Din Jamshīd al-Kāshī and Qādizāda al-Rumī. He is also known for his co