Significant Ottoman Mathematicians and their Works
Dr. Salim Ayduz*
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Table of contents
4.1. Qadizada al-Rumi (d. ca 1440)4.2. ‘Ala al-Din ‘Ali al-Qushji (ca 1402-1474)4.3. Khalil al-Husayni (15th century)4.4. Yusuf Sinan Pasha (d. 1486)4.5. Hajji Atmaca al-Katib (d. after 1494)4.6. Lutfullah al-Toqati (d. 1494)4.7. Mirim Celebi (d. 1525)4.8. Eliya Mizrahi (ca 1450-1526)4.9. Nasuh ‘Ali al-Silahi al-Matraki (d. 1564)4.10. Taqi al-Din ibn Ma'rûf (1520 - 1585)5. Ottoman mathematicians in the 17th and 18th centuries5.1. Khalil Fa'id Efendi5.2. As'ad Efendi al-Yanyawi (Yanyali Esad Efendi)5.3. Muhammad Istanbuli5.4. ‘Abd Al-Rahim Al-Mar'ashi Efendi (d. 1736)5.5. Mustafa Sidki b. Salih Kethüda (d. 1183/1769)5.6. Ibrahim of Aleppo (d. 1776)5.7. Sekerzada Sayyid Fayzullah Sarmad (d. 1787)5.8. Gelenbevi Ismail Efendi (1730-1790)5.9. Kalfazada/Halifezada Ismail Efendi (d. 1790)5.10. Huseyin Rifki Tamani (d. Madina, 1817)6. Some mathematicians of the 19th and early 20th centuries6.1. Hoca Ishak Efendi6.2. Ahmad Tawfik Efendi (1807-1869)6.3. Hüseyin Tevfik Pasha of Vidin (1832-1893)6.4. Salih Zeki (1845-1921)7. Conclusion8. References***
- 2. Mathematicians before the conquest of Constantinople
- 3. General overview of Ottoman mathematics
- 4. Ottoman mathematicians in the 15th and 16th centuries
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. **
2. Mathematicians before the conquest of Constantinople
Figure 1: An Ottoman miniature showing a game of Matrak invented by Nasuh ‘Ali al-Silahi al-Matraqi. 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-'ulum al-diniyya) in their totality, and famous religious scholars such as Dawud al-Qaysari (d. 1350) , Taj al-Din al-Garadi (d. c. 1360) and Ala al-Din 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, Qadizada's (d. 1432) two works on astronomy and mathematics Sharh al-Mulakhkhas fi al-Hay'a and Tuhfat al-Ra'is fi Sharh Ashkal al-Ta'sis 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 .
3. General overview of Ottoman mathematics
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 Saman Madrasas and the Enderun 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-Qushji (d. 1474) and Fathullah al-Shirwani's (d. 1486) two students of Qadizada al-Rumi (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 Tartib 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. Nasir al-Din al-Tusi'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 fi al-Hisab written by Ali al-Qushji and Khulasat al-Hisab written by Baha al-Din al-'Amili 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 .
Baha al-Din al-'Amili's Khulasat al-Hisab was a course book throughout the Ottoman school system from the 17th century. Kuyucaklizada Muhammed Atif (d. 1847) translated the book into Turkish with commentaries as Nihayat al-Albab fi Tarjamati Khulasat al-Hisab. 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 Khulasat al-Hisab. 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 Miftah al-Hisab (anonymous), Risala fi ilm-i Hisab (anonymous) and Miftah al-Mushkilat (Muhammed Musa-i Wafi). 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 qawa'id-i ‘ilm-i hisab 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-sadad written by Ali b. Veli Hamza b. al-Jazairi al-Maghribi (d. 1614) in 1590 and was presented to Sultan Murad 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 fi ‘ilm al-Hisab, al-Luma fi al-Hisab, al-Ma'una fi al-Hisab al-Hawa'l written by Ibn al-Ha'im (d. 1412) and Talkhis A'mal al-Hisab written by the Moroccan mathematician Ibn al-Banna (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-Mufassalat bi-'l-La'ali al-Nayyirat fi A'mali Zawat al-Asma wa-'l-Munfasilat written by Jamal Al-Din Muhammed b. Ahmad b. Muhammad b. Piri al-'Alwani, also known as Ibn Piri (d. 1631) ; Al-Mawahib al-Saniyya fi Ilm al-Jabr wa-'l-Muqabala and Sharh al-Yasminiya fi al-Jabr wa-'l-Muqabala written by Ibn Al-Jamal (d. 1662); al-Mustahzarat fi Hisab al-Majhulat by Kuyucaklizade Muhammed Atif (d. 1847) and Tuhfat al-Hisab 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 Sekerzada Feyzullah Sermed (d. 1787) translated the book entitled Maqsadayn fi 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-Jadawil al-Ansab by Gelenbevi Ismail Efendi in 1787; Logaritma Risalasi by Huseyin Rifki Tamani (d. 1817) and Logaritma Risalasi by Muftuzade Osman Saib (d. 1864) .
The Hendesehane-i Humayun (Royal Mathematical School) was the first institution that was designated for modern military technical education in the Ottoman State. The Hendesehane, 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 Muhendishane-i Humayûn (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 Gelenbevi Ismail Efendi (d. 1790) and Palabiyik Muhammad Efendi (d. 1804).
Figure 2, 3 & 4: Sample pages of Taqi al-Din ibn Ma'ruf's Hisab al-munanjjimin wa-'l-jabr wa-'l-muqabala. Source: Süleymaniye library in Istanbuly, Carullah collection, MS 1454.
4. Ottoman mathematicians in the 15th and 16th centuries
4.1. Qadizada al-Rumi (d. ca 1440)
His full name was Salah al-Din Musa ibn Muhammad ibn Mahmud Qadizada al-Bursawi al-Rumi. He was born in Bursa, Turkey, (hence his name al-Rumi, from the Arabic name al-Rum for the Byzantine and Ottoman States). His grandfather and father were judges/Qadi 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 Shah-i-zinda (Living King) in Samarkand.
Qadizada al-Rumi 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 fi'l-hay'a (Commentary on the ‘Compendium on Astronomy') and Sharh Ashkal al-Ta'sis (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 Risala fi Istikhraj Jaybi Daraja Wahida (Treatise on the Calculation of the Sine of a One Degree of the Arc).
Qadizada's two students ‘Ali al-Qushji (d. 1474) and Fathullah al-Shirwani (d. 1486) influenced Ottoman science by disseminating work on mathematics and astronomy. In the introduction to his Tuhfat al-Ra'is Fi Sharh Ashkal al-Ta'sis (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 fatwas in religious matters, the officials who run the affairs of state, and the qadis 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, Qadizada 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 : Qadizada wrote this commentary on Samarkandi's Ashqal 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 .
- Risala dar bayan-i istikhraj 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 Ghiyath al-Din al-Kashi. Although it is a commentary on the treatise of al-Kashi titled Risala al-watar wa'l-jayb (Treatise on Chord and Sine), due to the originality of the subject, al- Rumi is often regarded as the author of the treatise . According to Salih Zeki (d. 1921), it is the most important treatise by him .
- Risala fi al-misaha (Treatise on surveying), in Persian: In the prologue of the treatise, al-Rumi 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 (qaidah) .
4.2. ‘Ala al-Din ‘Ali al-Qushji (ca 1402-1474)
This scholar's full name is Qushci-zada Abu al-Qasim ‘Ala al-Din 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-zada" or "Qushji." He received advanced education from outstanding scholars such as Ulugh Beg, Giyath al-Din Jamshid al-Kashi and Qadizada al-Rumi. He is also known for his contributions to Ulugh Beg's Zij which was prepared on the basis of the observations conducted at Samarkand Observatory under the guidance of Ulugh Beg.
After Ulugh Beg's demise in 1449, he left Samarkand first for Herat, then Tabriz and finally Istanbul. While he was in Tabriz, Uzun Hasan  sent ‘Ali al-Qushji as an emissary to Muhammad II, who was impressed by him. Ali al-Qushji and Muhammad II had many scientific discussions. Sultan Muhammad was very pleased with him and asked him to remain in Istanbul permanently. Accepting the invitation, ‘Ali al-Qushji came to Istanbul with his family after the end of his duty as emissary (around 1472). Appearing before Sultan Muhammad, he presented the Sultan with the mathematical treatise al-Muhammadiyya, which was dedicated to him. Having been appointed as a Mudarris (teacher) of the Hagia Sophia school of Sultan Muhammad, ‘Ali al-Qushji spent the last couple of years of his life in Istanbul where he died in 1474.
A teacher of many students during his life, ‘Ali al-Qushji was a polymath scholar and a particular authority on astronomy and mathematics as well as many different disciplines such as language, religion, philosophy and mathematical sciences. He introduced new ways of understanding and exploring these disciplines and his works made an impact on scientific activities in both the Muslim and European worlds. He was instrumental in the importation of the Timurid Samarkand tradition to the Ottomans.
‘Ali al-Qushji was the author of many works on mathematics, astronomy, philosophy, and language, some of which were the outcome of original research, whilst others were textbooks for teaching or treatises focusing on specific problems. He wrote twelve works on mathematics and astronomy. One of them is his commentary in Persian on the Zij-i Ulugh Beg. His two works in Persian, namely, Risala fi al-Hay'a (Treatise on Astronomy) and Risala fi al-Hisab (Treatise on Arithmetic) were used as a course book in the Ottoman schools. He revised these two works in Arabic with some additions under new titles, al-Fathiyya (Commemoration of Conquest) and al-Muhammadiyya (The Book Dedicated to Sultan Muhammad), respectively. Both books won approval and were translated into other languages. Many commentaries were written about them. Altogether Ali al-Qushji wrote 32 books and treatises, although some of his works are regrettably lost . Copies of those works which have survived have made their way into modern-day libraries. His important works are listed below.
- Al-Risala al-Muhammadiyya fi al-hisab (Arabic): Treatise on Arithmetic. It was dedicated and presented to Sultan Muhammed II. Al-Qushji used the terms "muthbat" and "manfi" for added and subtracted quantities instead of the standard terms "za'id" and "naqis". Al-Qushji's terms are translations of Chinese terms and are presently used for positive and negative quantities in Iran, Turkey, Central Asia, and Azerbaijan; European terms for these quantities came from al-Qushji's terms through Byzantine mathematicians .
- Risala dar ilm-i hisab (Persian): Treatise on the Science of Arithmetic. Also known as Mizan al-hisab (Balance of Arithmetic) and Zubdat al-hisab (Essence of Arithmetic). It contains three books: Indian arithmetic, sexagesimal fractions, and geometry.
- Risala fi al-qawa'id al-hisabiyya wa'l-dala'il al-handasiyya: Treatise on Arithmetic Rules and Geometric Indications . It is a treatise on the sine of 1o.
- Risala dar hisab u handasa (Persian): Treatise on Arithmetic and Geometry.
- Risala fi istikhraj maqadir al-zawaya min maqadir al-adla' fi al-muthallathat al-ghayr qa'imat al-zawaya al-haditha min qisiyy al-dawa'ir al-'izam: Treatise on Determining the Magnitudes of Angles of a Triangle by the Magnitudes of Sides in Non-Rectangular Triangles Consisting of Arcs of Great Circles [of a Sphere]. (Suleymaniye Library, Carullah collection MS 2060).
4.3. Khalil al-Husayni (15th century)
Khayr al-Din Abu ‘Abdullah Khalil ibn Ibrahim al-Husayni was another mathematician, who worked in Istanbul at the court of Muhammad II . There is very limited information about his biography. Two of his significant works on mathematics have reached today. His works are mainly on the subject of accounting mathematics.
- Miftah-i kunuz-i arbab-i qalam wa misbah-i rumuz-i ashab-i raqam: (Persian) (Key to Treasures of the Masters of the Pen and Lamp of Symbols of Rulers of Figures). It is also known as Risala fi al-hisab (Treatise on arithmetic). The Treatise is dedicated to Muhammad II and divided into an introduction (muqaddimah), ten chapters and one epilogue (khatimah). Chapters 1-4 deal with different kinds of multiplication, 5-6 are devoted to different kinds of division, chapter 7 is on problems, 8-10 are dedicated to extraction of roots of 2nd, 3rd, and 4th powers . This treatise was very well known and circulated among the state (diwan) accountants. It was compiled for their daily use for accounting calculations. The name of Muhammad II was cited in the introduction (muqaddima). It was translated into Turkish by Khalil al-Husayni's student Pir Mahmud Sidki al-Edirnevi. The section on the khata'ayn rule (double mistake) was translated into Turkish by Muhyi al-Din Hajji Atmaca al-Katib .
- Mushkil gusha-yi hisab u mu'dil numa-yi kitab (Book of Difficulties in Arithmetic Solutions and those that are Incomprehensible), in Persian. Also known as Mukhtasar fi al-hisab (Concise [Book] on Arithmetic , it was dedicated to the Sultan Beyazid II (1481-1512) and divided into an introduction, six sections and an epilogue (khatimah) .
- Risala-i Jabr u-Muqabala (Persian). It was not cited by any sources and there is only one copy at Nuruosmaniye Library, MSS 2980/2 .
4.4. Yusuf Sinan Pasha (d. 1486)
Sinan al-Din Yusuf ibn Khidr Beg ibn Jalal al-Din (d. 1486), known by the names "Sinan-Pasha" and "Khwaja Pasha", was the vizier of Sultan Muhammad II; he worked in Istanbul and Edirne. He was a well known historian, theologian, mathematician and astronomer .
He has only one work on mathematics: Risala fi Bayani Mas'alatin Handasiyya or Risala fi Istihraci Zaviye Hadde ‘Iza Furida Harakatu Ahadi or Risala fi al-munfarija ta'siru hadda qabla an ta'sira qa'ima; (Treatise that Obtuse (Angle) can become Acute without being Right) . Ihsan Fazlioglu worked on this short treatise and published it with a lengthy analysis .
4.5. Hajji Atmaca al-Katib (d. after 1494)
Muhyi al-Din al-Hajji Muhammad ibn al-Hajji Atmaja al-Katib was a mathematician of the 15-16th centuries . We have no information about his life. Most probably, he was one of the accountants for the State Diwan . He lived during the reigns of Muhammad II and Bayezid II.
He wrote two works on arithmetic:
- Majma'-i qawa'id-i ‘ilm-i hisab/Jami' al-qawa'id (Collection of rules of the science of arithmetic), in Turkish. It was completed in 1494 and dedicated to the Sultan Bayezid II. It is an account book which was composed for scribes and accountants who were working for the administration. In the epilogue, he explains why he prepared this book saying that in the state administration, accountancy apprentices need to learn accounting mathematics through this book. He also indicates that most of the books on this subject are written in either Arabic and Persian, hence new staff need a book in Turkish in this subject to understand it easily . The book contains three chapters and a prologue (tatimma): the first chapter is about integers; the second on calculations with rational numbers and in the third chapter there are forty problems with solutions. It was very commonly used among court scribes until the last centuries of the Ottoman state .
- ‘Ilm al-hisab (Science of Arithmetic) .
- He also translated the sixteenth chapter of Khayr al-Din's book Miftah‘s as an independent treatise with the title Tarjamat al-fasl al-sadith ‘ashara fi bayan al-khata'ayn min miftah-i kunûz-i arbab-i kalam wa misbah-i rumûz-i ashab-i raqam .
4.6. Lutfullah al-Toqati (d. 1494)
His full name is Lutfullah Muhammad bin Hasan al-Toqati, also known as Molla Lutfi (d. 1494). He was born in Tokat (Turkey). There is very limited information about his life; he studied mathematics under Sinan Pasha and later advanced studies with Ali al-Qushji. When Sinan Pasha was appointed as a Grand Vizier to the Sultan Muhammad II, Molla Lutfi was also appointed as the librarian of the Sultan in his personal library at the palace. When Sinan Pasha was exiled to Sivrihisar, Molla Lutfi was also removed from his post and accompanied Sinan Pasha. When Bayezid II ascended to the throne, they were pardoned and Molla Lutfi was appointed to a madrasa in Bursa as a mudarris. Later he moved to Edirne as a mudarris (teacher), returned to Bursa, then went back to Istanbul, and was appointed among the teaching stadff of Sahn Saman schools. Due to his extremist and liberal ideas on Islamic matters, he was accused as a heretic and was executed in January 1495 . He wrote a treatise about the classification of sciences in Arabic titled Mawdu'at al-Ulum (The subjects of sciences).
In mathematics, he compiled in Arabic Risala fi tad'if al-madhbah (Treatise on Duplication of the Altar). It is a geometry book about a problem known as the Delos Problem . Some of the book was written by him and the rest is a translation and compilation of other books on the subject .
4.7. Mirim Celebi (d. 1525)
Mahmud ibn Muhammad ibn Qadizada al-Rumi, known as "Mirim Çelebi", was the grandson of Qadizada al-Rumi (from his son) and of Ali al-Qushji (from his daughter). Born in Samarkand, he worked in Gallipoli, Edirne, and Bursa and died in Edirne in 1525 . He studied mathematics under Ali al-Qushji and was educated in various schools in Istanbul .
Mirim Celebi was a well-known astronomer and mathematician of his time. He made great contributions to the establishment of the Ottoman scientific traditions of mathematics and astronomy and was known for the commentary he wrote on the Zij of Ulugh Beg and his treatises on astronomy. He composed books on mathematics and cosmology.
His book in Persian Dastûr al-'amal wa-tashih al-Jadwal (Rules of operations and correction of Tables) is a commentary on Ulugh Beg's Zij. A treatise with the same title containing an exposition of the treatise (Risala al-watar wa'l-jayb) of al-Kashi was written by al-Rumi's grandson Mirim Chelebi. This treatise is about astronomy, but the first chapter dealings with mathematics. Risala (A Treatise): Trigonometrical part of Dastûr al-'amal wa tashih al-jadwal (Rules of Actions and Corrections of the Table) this part of the treatise containing exposition of determining sine 1o according to the works of Qadizada al-Rumi and al-Qushji . According to Woepcke, Mirim Celebi studied the values of trigonometric descriptions and obtained some original results .
4.8. Eliya Mizrahi (ca 1450-1526)
Eliya Mizrahi was born and lived in Istanbul under Sultans Muhammad II, Bayezid II (1481-1512), Selim I (1512-1520) and Suleyman I (1520-1566). He was a Jewish scholar, a descendant of Byzantine Jews (Romaniot). He possessed the highest rabbinical authority of his time and was Chief Rabbi in the Ottoman State from 1498 onwards, and also a mathematician, astronomer, physicist, and philosopher .
His mathematical writings include:
- Sefer ha-mispar (Book of Number). Mizrahi learned decimal fractions from the Istanbul mathematicians and was a link between them and the mathematicians of Western Europe.
- Commentary on Euclid's "Elements" .
4.9. Nasuh ‘Ali al-Silahi al-Matraki (d. 1564)
Nasuh b. Karagöz al-Bosnawi or Nasuh b. Abdullah al-Silahi al-Matraki, or "Matrakçi Nasuh". His family originated in Bosnia. His father or grandfather was drafted into the state service. He was a court official and renowned in the 16th century as a mathematician, historian, geographer, cartographer, topographer, musketeer, and was an outstanding knight, calligrapher and engineer. Because he was a musketeer, he was also called al-Silahi (the musketeer or gunman). He was a polymath thinker, writer, and artist who pioneered a particular artistic style for depicting cities. He wrote books in these fields, all in Turkish.
He received the nickname "Matrakçi" after he created the game called Matrak, which means ‘amazing' in Turkish (with ‘çi' as a suffix). Therefore his nickname means "who plays (invents) the amazing game" .
Matrakçi Nasuh was educated and trained in the Palace school Enderun during the reign of the Bayezid II and studied with Sai Çelebi, one of Sultan Bayezid II's teachers. During the reign of Sultan Selim I, he started to distinguish himself as a knight. He went to Egypt in 1520 for advanced studies and attended military games, at which he became unrivalled. He was given a decree on war games indicating his outstanding talent.
In the field of mathematics, Al-Matraqi wrote two books in Turkish with the purpose of facilitating the work of clerks of the state council (Divan katipleri) and the state accountants (muhasebeciler). These two books are important in tracing the development of Turkish as a language to a level where it was suitable for use as a mathematical language. They are also important in following the history of the Ottoman solution of accountant's mathematical problems. It is the second most important book after Hajji Atmaca's work in this field.
A brief discussion of his mathematical books follows.
- Jamal al-Kuttab wa Kamal al-Hussab (Beauty of Reckoners in the Perfectness of arithmetic). Al-Matraqi wrote his first book in 1517 and dedicated it to Sultan Selim I. Jamal al-Kuttab included two chapters. The first one is about Indian numerals, mathematical operations, fractions, scales, and measurements. Although he says that the second chapter is devoted to "miscellaneous matters", it is no extant in the surviving manuscripts .
- ‘Umdat al-Hussab or Hisab fi furud al-muqaddar (Support of arithmetic in propositions of all magnitudes). This is his second book, written in 1533. ‘Umdat al-Hussab is an expanded version of the Jamal al-Kuttab. The title of the first chapter is "miscellaneous subjects"; it has twenty-two sections (fasl): 1) siyaqat figures, 2) Indian figures, 3) addition of integers, 4) algebra and fractions, 5-6) duplication and mediation, 7-8) application of fractions in craft and trade, 9-11) multiplication and division of integers and fractions, 12-15) measures of length, volume, and weight, 16) drawings, 17) proportions, 18) taxes, 19) rule of "two errors", 20) addition of fractions, 21) double-false, 22) additions of fractions (jam' kusur ma'a kusur) .
The second chapter is entitled "solution of the 50 problems". Some figures and diagrams were added in this version. In addition to the subjects mentioned, this book also contains weights, measurements (zira, endaze, kilajat, qantar, misqal, dirham), ratio, division with proportion and geometric methods, all essential for accountants. After every subject, Al-Matraqi gives examples offering new measurement divisions which were previously unknown. In the first part, the six fundamental operations of classical arithmetic are extensively investigated for positive integers and rational numbers. In addition, the "double-false" rule used to find an exact solution for a linear equation is analysed. In the second part, several issues are explored. According to Al-Matraqi, these issues were rarely mentioned in previous works; but accountants should definitely learn them. The book deals with various other subjects, such as inheritance and tax, essential to accountants, and which are studied through examples of calculations. When Al-Matraqi wrote the second book, the first one had been almost forgotten. While we have about fifteen copies of the second book, only four copies of the first one survive. This indicates how this second book was well used by accountants.
4.10. Taqi al-Din ibn Ma'rûf (1520 - 1585)
Taqi al-Din ibn Ma'ruf was a major Ottoman scientist in the second half of the 16th century. From 1571, he settled in Istanbul, the capital of the Ottoman State and excelled in several scientific fields such as mathematics, astronomy, engineering, mechanics, and optics. The greatest astronomer of the period, Taqi al-Din combined the Egypt–Damascus and Samarkand traditions. He wrote more than thirty books in Arabic and Turkish on the subjects of mathematics, astronomy, mechanics, and medicine.
Taqi al-Din was born in Damascus and he completed his education there. He moved from Egypt to Istanbul for the third time in 1570. He was respected and appreciated by Hoja Sa'ad al-Din Efendi (d. 1599), the tutor of Sultan Murad III (1574–1595). In 1571, he was appointed as munajjimbashi (chief astronomer) by Sultan Selim II (1566–1574). Shortly after Sultan Murad III's accession to the throne, he started the construction of the Istanbul observatory under the patronage of the Sultan. It is understood from his Zij titled Sidrat Muntaha al-Afkar (The Nabk Tree of the Extremity of Thoughts) that he made observations in the year 1573. It is generally agreed that his famous observatory was demolished on 22 January 1580. Therefore, it can be estimated that he carried out observations from 1573 until 1580.
In addition to the existing instruments of observation, Taqi al-Din invented new ones such as the Mushabbaha bi'l-manatiq (sextant) and Dhat al-awtar in order to determine the equinoxes. Moreover, he also used mechanical clocks in his observations. Taqi al-Din developed a different method of calculation to determine the latitudes and longitudes of stars by using Venus and the two stars near the ecliptic .
Starting with Ptolemy in the 2nd century CE and continuing until Copernicus in the 16th century, the Western world used chords for measuring angles. For this reason, the calculation of the value of the chord of 1o has been an important matter for astronomers. Thus, while Copernicus used the method based on the calculation of the chord of 2o that yielded an approximate value, Taqi al-Din used trigonometric functions such as the sine, cosine, tangent, and cotangent to measure the values of angles, in line with the tradition of Islamic astronomy. Inspired by Ulugh Beg, Taqi al-Din developed a different method to calculate the sine of 1º. Furthermore, he applied decimal fractions, which had been previously developed by Islamic mathematicians such as al-Uqlidisi and al-Kashi, to astronomy and trigonometry, prepared sine and tangent tables accordingly, and used them in his work titled Jaridat al-Durar wa Kharidat al-Fikar.
Taqi al-Din wrote many books about mathematics and astronomy. Here are his works on mathematics:
- Kitab al-nisab al-mutashakkila fi al-jabr wa al-muqabala (Book on coinciding ratios in algebra) : It was divided into a prologue, three sections and an epilogue.
- Bughyat al-tullab fi ‘ilm al-hisab (Aim of Pupils in the Science of Arithmetic) : It is enclosed also in Al-Hisab al-hindi, a handbook which contains the book Hisab al-muanjjimin wa-'l-jabr wa al-muqabala . The codex had three chapters: 1) on arithmetic with decimal figures, 2) on arithmetic with hexadecimal figures, 3) on algebra.
- Kitab tastih al-ukar (Book on Projecting Spheres onto a Plane) = Dastur al-tarjih fi qawa'id al-tastih (Preferred Rule in Foundations of Projecting on a Plane) or Tahrir Kitab al-ukar li-Thawudhusiyus (Exposition of "Book on Spheres" of Theodosius) . It is mentioned under the first title in Kashf al-Zunun of Hajji Khalifa (II, 288; III, 226). It is a treatise on stereographic projection which could be part of a larger astronomical work. The book is dedicated to Hojja Sa'ad al-Din Efendi and has two chapters.
- Risala fi tahqiqi ma qalahu ‘l-'alim Giyath Jamshid fi bayani ‘l-nisba bayna ‘l-muhit wa-'l-qutr. Taqi al-Din discusses here the ideas of Giyath Jamshid al-Kashi's book al-Risalat al-muhitiyya .
 R. Hillenbrand, "Madrasa", The Encyclopaedia of Islam, New Edition, Leiden: E. J. Brill, 2000, V, 1144.
 Ihsan Fazlioglu, "Davud Kayseri", Yasamlari ve Yapitlariyla Osmanlilar Ansiklopedisi, Istanbul: Yapi Kredi Yayinlari, 1999, I, 370-371.
 Katip Celebi (d. 1658) claims that, in his time, the interest in rational sciences decreased, and that in time they were excluded from the madrasas teaching. However, his viewpoint should be re-examined in the light of historical facts and in the broader context of the development of the cultural and intellectual life in the Ottoman capital, especially in the seventeenth century. The ample evidence we have in the rich Ottoman scientific literature surveys published by IRCICA indicates a progressive curve in the inclusion of the rational sciences and does not confirm Katip Celebi's statement. For a critical evaluation of Katip Celebi's words, see Ihsanoglu (1996, p. 39-84) and Ihsanoglu (2004).
 For the copies of the first book see OALT 1. pp. 9-21; for the second book OMLT I, pp. 7-18.
 Cevat lzgi, "Osmanli Medreslerinde Aritmelik ve Cebir Egitimi ve Okutulan Kitaplar", Osmanli Bilimi Arastirmalari, no. 3401, (1995).
 This book was printed in Roma: Kitab Tahrir Usul li-Uqlidis min ta'lif khwaja Nasir al-Din al-Tusi. Euclidis elementorum geometricorum libri tredecim. Ex traditione doctissimi Nasiridini Tusini. Nunc primum Arabice impressi. Romae: in Typographia Medicea, 1594.
 M. L'Abbè Toderini, de la Littérature des Turcs, Traduit de l'Italien en Francois par Tournant, trs. M. L'abbe De Cournand, vol. I, (Paris: Poincot, 1789), 100-105.
 Suleymaniye Library, Haci Mahmud, MS 5721. Kandilli Rasathânesi Library, MS. 127/2. O. R. Kahhala, al-Mustadrak alâ Mu'cam al-muallifîn, Beirut 1985, p. 673; Cevat Izgi, Osmanli Medreselerinde Ilim, Istanbul 1997, I, 223, 409-410, 445-446: Ihsan Fazlioglu, "Hesab", Diyanet Islam Ansiklopedisi, Istanbul 1998, XVII, p. 251.
 For the list of the copies of the MS see: OMLT, I, 290-293.
 Ihsan Fazlioglu, "Cebir", Diyanet Islam Ansiklopedisi, Istanbul 1993, VII, 199.
 Although it is not an algebra book, it contains some problems related to algebra. OMLT, I, 132-134.
 Ekmeleddin Ihsanoglu, "Some Remarks on Ottoman Science and its Relation with European Science & Technology up to the End of the Eighteenth Century," Papers of the First Conference on the Transfer of Science and Technology between Europe and Asia since Vasco da Gama (1498–1998). Journal of the Japan–Netherlands Institute 3: 45–73, 1991.
 Frédérik Hitzél, "Défense de la Place Turque d'Oczakow par un Officier du Génie Francaise (1787)", in Ikinci Tarih Boyunca Karadeniz Kongresi Bildirileri, ed. Mehmet Saglam (Samsun 1990), 639-655.
 Rosenfeld, p. 378.
 Rosenfeld, no 808, M2.
 OMLT, I, 6-18.
 Rosenfeld, no 808, M3; no. 802, M4.
 OMLT, I, 3-5.
 Suleymaniye Library, Esad Efendi, MS 2023/2, folios 35a-43a; OMLT, I, 5.
 Uzun Hasan or Hassan (1423 – 1478), Sultan of the Aq Qoyunlu dynasty, or White Sheep Turkomans. Hassan ruled between 1453 and 1478.
 A. Süheyl Ünver, Ali Kusçu Hayati ve Eserleri, Istanbul University, Faculty of Science, Monograph, no. 1, (1948).
 Rosenfeld, p. 286.
 Petersburg, no. A. 134/2.
 Rosenfeld, p. 287.
 Rosenfeld, p. 287.
 To see the copies of the MSS see: OMLT, I, 34-35.
 Rosenfeld, p. 288.
 To see the copies of the MSS see: OMLT, I, 35.
 OMLT, I, 36.
 Rosenfeld, p. 290 (nr. 858): Hoca Sadettin, Tâc al-tawârih, Istanbul 1280, II, pp. 498-500; Hajji Khalifa; Kashf al-zunun, II, pp. 1819, 1236, 1311; Adnan Adivar, Osmanli Türklerinde Ilim, Istanbul 1982, pp. 49-50; Ibn al-Imâd, Shazarât al-zahab, Cairo 1350, VII, pp. 351-352; C. Woodhead, "Sinan Pasha, Khodja", EI2, s. 630-631; Cevat Izgi, Osmanli Medreselerinde Ilim, I, s. 299, 385, 403.
 Istanbul, Koprulu Library, Mehmed Asim Bey, MSS 721. Ihsan Fazlioglu, "Ali Kusçu'nun bir hendese problemi ve Sinan Pasa'ya nispetedilen cevabi, Tenkidli metin ve çalisma", Divan, 1996, I, 85–106. OMLT, I, 27-28.
 See Sinâneddin Yusuf Efendi (Sinan Pasa) (891/1486) (2006).
 KZ, V, 404; OM, III, 252; MAMS, II, 535-536; OMLT, 29-31; SSM, 170.
 Rosenfeld, p. 310 (nr. 918).
 Marmara University, Theology Faculty Library, Genel Yazmalar nr. 185, folios 1b-2a; Suleymaniye Library, Esad Efendi MSS 3176.
 OMLT, I, 29-31; M. K. Özergin, "Haci Atmacagolu ve Eseri", Islam Düsüncesi, V, Istanbul 1968, 312-316.
 Budapest Török MS 0177.
 OMLT, I, 31.
 OMLT, I, 37-40.
 Suleymaniye Library, Esad Efendi 3596/1. OMLT, I, 37-39.
 Hajji Khalifa, Kashf al-zunûn, II, 1715, 1716.
 Rosenfeld, no. 940.
 Ihsan Fazlioglu, "Mirim Çelebî", Diyanet Ýslâm Ansiklopedisi, Ýstanbul 2005, XXX, 160-161.
 Rosenfeld, no. 940, M1.
 Franz Woepcke, "Discussion de deux méthodes arabes pour déterminer une valeur approchée de", in Études sur les mathémateques arabo-islamiques, ed. by Fuad Sezgin, Frankfurt 1986, pp. 614-638.
 Rosendfeld, no. 943; E. Wiedemann, "Über physikalische Aufgaben bei Elia Misrachi", in Monatsschrift für Geschichte und Wissenschaft des Judentums (Breslau), 54 (1910), Heft 2, pp. 224-232; (208), I, 434-442.
 Rosendfeld, no. 943.
 Salim Ayduz, "Nasuh Al-Matraki, A Noteworthy Ottoman Artist-Mathematician of the Sixteenth Century".
 Rosenfeld, no. 1001 M1; OMLT, I, 69.
 Rosenfeld, no. 1001 M2; OMLT, I, 70-73.
 On the work of Taqi al-Din, see the special section devoted to him on www.MuslimHeritage.com: Taqi Al-Din: Astronomy, Mathematics, Optics and Technology.
 Cairo (Miqat 557/3, 4 f., Taymur Riyada. 140/10), Oxford (I 88/3).
 Cairo (Riyada. 1023), Rome (Vatican Sbath 496/2). It is quoted in Qamus al-Riyadhiyyat of Salih Zeki (vol. II, p. 59).
 Süleymaniye library, Carullah, MS 1454, 55 folios.
 Cairo (Tal'at miqat 135 - anonymous), Istanbul, MS Kandilli 415/5, 12 folios.
 Istanbul, Kandilli, nr. 208/8, 5 f.
*Lecturer at Fatih University, Istanbul and Senior Researcher at the Foundation for Science, Technology and Civilisation (FSTC), UK.
**I am grateful to Prof. Ihsan Fazlioglu for his valuable contributions and comments on this article.
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by: FSTC Limited, Mon 19 December, 2011