Since the Ottoman early traditional period, arithmetical books that could improve bookkeepers’ mathematics had already begun to be written. One of them is Pir Mahmud Sıdkı Edirnevi’s translation of Miftah-i Kunuz-i Arbab al-Kalam va Misbah-i Rumuz-i Ashab al-Rakam, which belongs to the 15th century and was written in Persian owing to the Persian effects on the financial foundation in 1505. The attention paid to arithmetic in Edirnevi’s Terceme-i Miftah-i Kunuz (Ilm-i Arkam-i Taksimat) is noticed in the chapters on root extraction, and it differs with approximation methods from other mathematical books in its century. Thus, our study mentions Edirnevi’s approximation methods on 3rd and 4th degree root extractions and intends to define this translation’s contribution to Turkish mathematic compilations. In this respect, our study aims to reveal the pioneer position of Edirnevi in the reception of numeric analysis subjects in Ottomans through the unique and complete manuscript of Terceme-i Miftah-i Kunuz (Ilm-i Arkam-i Taksimat). Also, in our study, both the historical and the mathematical analysis are followed in a methodological way. Our study consists of three main parts which are related firstly to the historical background, secondly to the introduction of the book and thirdly to the mathematical analysis of texts on the approximation of root extraction in this book, and it includes an evaluation of Edirnevi’s role in this context. 

In Mesopotamian mathematics, the fact that the exact and approximate results were separated obviously could be inferred from the particular terms that point out this separation. Also, since the formulas used for root extractions look like Greek mathematicians such as Archytas (BC 345) and Hero of Alexandria (AD. ~60), it is possible to say that Mesopotamian mathematics had a significant effect on Greek mathematics and represents the first steps of the history of numeric analysis. In the late medieval periods, it is outstanding that Chinese mathematicians made the solutions of the polynomials possible through the root extraction methods, and this method resembled what William Horner invented for algebraic equations in the 19th century. 

The tendency that provided mathematical improvement in the medieval Islamic world and the Ottomans was a numeric perspective that had been developed as an alternative to a geometric perspective. It should be considered that Newton’s and Descartes’ revolutional attempts in their books occurred thanks to this numeric perspective. Although the revolutional process did not occur in the East as the West achieved in the early modern period, the results of this perspective in the East is meaningful for mathematical precision. For instance, there was more than one way to find the square root in Khwarizmi’s (d. ~ 847) Hindî reckoning book. Then, Abu’l-Hasan al-Uqlidisi (d. 980) proposed different rules and found more precise results, but Abd al-Qahir al-Baghdadi’s (d. 1037) rules were begun to be followed as a conventional method (approximation) since his result of approximate cube root was more precise. Subsequently, Ibn al-Haytham (d. 1040) noted the irrational roots with justifications, and mathematicians after the 11th century attempted to find higher degree roots. The formulas that were applied by Nasir al-Din al-Tusi (d. 1274) and Nizam al-Din Nisaburi (d. 1328) were not different from the one that is known as the Ruffini-Horner method of the 19th century. The conventional method (approximation) was initially promoted by Samaw’al al-Maghribi (d.~1175) through nth degree root extraction to the base-60 numeral system (sexagesimal), then by Jamshid al-Kashi (d. 1429) to the base-10 numeral system (decimal). Thus, in Kashi’s book named Miftah al-Hisab, both predecessors’ methods were combined and extracting the approximate root was generalized in any degree. Through using Miftah al-Hisab, which is of East origin in Ottoman madrasahs and making commentaries and copies of the concise texts of Ibn al-Banna’s Talkhis Amal al-Hisab, which is of Andalus origin, the Ottomans became aware of many numeric analysis methods inheriting various mathematical traditions in different geographies.

As mentioning root extraction methods in general reckoning books got usual in Ottomans, there were exact and approximate root extractions in high degree (fourth and fifth degree) in Ali al-Qushji’s (ö. 1474) or Ibn Hamza al-Maghribi’s (d. 1614) mathematical book. In addition, the Ottomans became familiar with these methods in not only integers but also fractional numbers through teaching Nisaburi’s mathematical textbook in madrasahs. Although the Ottomans had difficulties in applying these methods to the solution of highdegree equations, this subject was handled as a detached chapter in reckoning books with algebraic contents.

Findings of our study indicate that Edirnevi’s translation, which is an arithmetical book in principle, is based on calculation with measures and common fractional numbers, proportion, false position method, shares of dept claims and root extractions. Since finding rational roots were mentioned quite detailed in this book, it is avoided to give examples of large numbers’ irrational roots. Thus, after the integer part of the root is obtained at once, the calculation of the approximate part of the root is left to be paid attention. Because of this, Edirnevi did not need the tables that he describes the steps of the finding rational roots. Edirnevi followed the method known as conventional approximation in the medieval Islamic world in finding the approximate cube root. However, he could not succeed in finding the fourth degree root exactly. Although he did not fail in extracting the integer part of the root, the approximate part of the root is not precise enough. That is a serious disadvantage of his book. On the contrary, the fact that this book addressed Ottoman bookkeepers is considered, these are pioneer and unique enterprises among the mathematical books written by bookkeepers. Despite the absence of chapters about algebra in this book, it could be a reference for algebra teaching thanks to the close relation between the root extraction methods and the solution of equations. 

In conclusion, thanks to Edirnevi’s efforts, the most important methods in numeric analysis were integrated with Turkish mathematics compilation at the beginning of the Ottomans’ 16th century. That is also a rare and inspiring fact among the bookkeepers’ texts. Therefore, the pros and cons of Edirnevi’s approaches are important components of the mathematical precision in the Ottomans’ numeric analysis methods.