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This work is an elaboration of the commentary written by the Egyptian mathematician Sibṭ al-Māridīnī—i.e., a commentary on another commentary—on the *urjūzah* (versified introduction) to the science of algebra, originally composed by the Berber mathematician and man of letters Abū Muḥammad ‘Abd-Allāh al-Ishbīlī al-Marrakushī, also known as Ibn al-Yāsamīn, who died in 1204 (600 AH). Al-Yāsamīn summarized his mathematical knowledge in a versified treatise known as the Yāsamīnīyya (The treatise by al-Yāsamīn). Around the end of the 15th century, al-Yāsamīn’s verses were the object of a ...

This work is a printed edition of *Kita**̄b taḥri**̄r uṣu**̄l li-Uqli**̄dus* (The recension of Euclid's *Elements*) by one of the intellectual luminaries of the Islamic world, the Persian polymath Nasir al-Din Muhammad ibn Muhammad al-Tusi (1201–74). After his death al-Tusi was referred to as *al-muʿallim al-thalith* (the third teacher, with Aristotle and Farabi referred to as the first and second teachers, respectively). An extraordinarily prolific author, al-Tusi made notable contributions to most of the intellectual fields of his era, writing on theology, mysticism, logic ...

This treatise deals specifically with basic arithmetic, as needed for computing the division of inheritance according to Islamic law. It contains 48 folios and is divided into an introduction, three chapters, and a conclusion. The introduction discusses the idea of numbers as an introduction to the science of arithmetic. Chapter I discusses the multiplication of integers. Chapter II is on the division of integers and the computation of common factors. Chapter III deals extensively with fractions and arithmetic operations on them. The author, an Egyptian jurist and mathematician, was the ...

This work is by Abd-Allāh Ibn Bahā al-Din Muhammad Ibn Abd-Allāh al-Shanshāri al-Shāfīī, an expert in calculating al-Fardī (inheritance portions). The cover page of the manuscript bears a magical form or talisman for finding a lost object. The main text is a detailed commentary on *Tuhfat al-ahbāb fi al-hisāb* (The friendly gift of arithmetic) by the renowned Egyptian scholar Badr al-Dīn Muhammad Ibn Muhammad Ibn Ahmad (1423–1506), known as the Sibt (grandson of) al-Mardini, who taught arithmetic and astronomy in Alazhar for several years. The original work has an ...

This work is an elaboration of the commentary written by the Egyptian mathematician Sibṭ al-Māridīnī (i.e., a commentary on another commentary), on the versified introduction, or *urjūzah*, to the science of algebra, originally composed by the Berber mathematician and man of letters Abū Muḥammad ‘Abd-Allāh al-Ishbīlī al-Marrakushī, also known as Ibn al-Yāsamīn (died 1204 [600 AH]). Ibn al-Yāsamīn’s work has not been examined in detail by scholars, so the apparent inclusion in this treatise of original lines by Ibn Yasamīn is of great importance in studying his contribution ...

This manuscript is a copy of *al-Risāla al-Fatḥīya fī al-a‘māl al-jaybīya* (The introductory epistle on sinusoidal operations) by Muḥammad ibn Muḥammad ibn Aḥmad Abu ‘Abd Allāh, Badr al-Dīn (1423–1506), known as Sibṭ al-Māridīnī or the grandson of al-Māridīnī, in honor of his mother’s father, a famous astronomer. The manuscript consists of 16 pages of 14 lines each, and includes an introduction and 20 *bābs* (chapters or articles). They range in length from a few lines to a page, and cover such topics as determination of the cardinal ...

The establishment of the Berber-Muslim dynasty of the Almohads in North Africa and Andalusia in the 12th century coincided with the decline in scientific advances in many fields of knowledge, including medicine. This was not the case with mathematics, and the treatise preserved in this manuscript together with other works by the same author stand as clear proof of the liveliness of this field under the rule of the Almohads and of the Marinid dynasty that followed. Abū ‛Abbās Ahmad Ibn al-Bannā was born in the second half of the ...

This manuscript offers a clear example of the liveliness of the North African mathematical tradition under the Muslim-Berber dynasties that ruled over the Islamic West from the 12th century to the first half of the 17th century. They were the Almohads (12th–13th centuries), the Marinids (13th–15th centuries), the Wattasids (15th–16th centuries), and the Saadis (16th–17th centuries). While there was little scientific advance in other fields in this period, the mathematical sciences kept on developing, as reflected both in the composition of original works and in commentaries ...

The mathematical tradition that developed in North Africa during the Middle Ages continued to attract the interest of scholars in subsequent centuries. Medieval treatises were extensively read and made the subject of commentaries. In many cases, these commentaries became the object of other works—or supercommentaries—aimed at further clarifying the subject of the original treatises. This manuscript is an example of this phenomenon. In the 12th century, the North African mathematician ‘Abdallāh ibn Hajjāj ibn al-Yāsamīn summarized his mathematical knowledge in a versified treatise known as Yāsamīnīyya (The treatise ...

This manuscript by Badr al-Dīn Muhammad ibn Muhammad ibn Ahmad ibn Muhammad ibn al-Ġazal (1423–1506) contains a commentary on, and abridgement of, the astrological treatise on the calculation of the movement of stars and planets, *Kašf al-haqā’iq fī hisāb al-daraj wa-al-daqā’iq* (The uncovering of the facts regarding the calculation of degrees and minutes), by the Egyptian astronomer and mathematician Ahmad ibn Rağab ibn al-Mağdī (1366–1447). Ibn al-Mağdī was a disciple of the famous ‘Abdallāh al-Māridīnī (or al-Mārdīnī), who was the grandfather of the author of this ...

The author of this mathematical treatise, Bahā' al-Dīn Al-‘Amilī (1547–1621), is considered one of the leading intellectuals of 17th-century Safavid Persia (present-day Iran). He was born in Baalbek (present-day Lebanon) but moved to Persia in his youth where he devoted his entire life to study. He excelled in various fields, leaving a legacy of more than 80 books on a wide variety of subjects that included theology and mysticism, astronomy, mathematics, poetry, and architecture. He wrote in both Persian and Arabic. He was the teacher of Mulla Sadra ...

The treatise in this manuscript is a commentary on a mathematical treatise by Šihāb al-Dīn Aḥmad ibn Muḥammad Ibn al-Hā’im (circa 1355–1412). Ibn al-Hā’im taught mathematics and Islamic jurisprudence, subjects on which he wrote extensively. The erudite Badr al-Dīn Muhammad Sibt al-Māridīnī (circa 1423–1506), who was at the time working as *muwaqqit* (timekeeper) at the Al-Azhar mosque in Cairo, composed this short commentary less then 60 years after the death of Ibn al-Hā’im. Following widespread tradition in Islamic lands, Sibt al-Māridīnī included in the title ...

This manuscript is a commentary on the treatise *Al-Tuffāḥa fi ‘ilm al-Misāḥa* (The apple in the science of measurement), which was written at the beginning of the 12th century by the mathematician Aḥmad ibn Muḥammad al-Ašh‘ari. The study of measures and measurement techniques (*‘ilm al-misāha*) was of great interest to Arabic mathematicians during the Middle Ages, both from theoretical and practical points of view. The ability to calculate the dimensions of landholdings was extremely important when it came to determining the correct amounts for inheritances and to calculating taxes ...

Islamic law goes into great detail on the subject of the division of inheritances (*farā'id*) among heirs. For this reason, inheritances have received extensive treatment in books of *fiqh* (Islamic law) and been a subject of study for mathematicians as well. *Qabas al-Daw' fī al-Hisāb* (The illumination of inheritance calculation) was copied by its author, ‘Abd al-Raḥman ibn Aḥmad ibn 'Ali al-Ḥamidi, in this 1589 manuscript. The work, which he dedicated to the son of the *Šāf‘ī* jurist Šams al-Dīn Muhammad al-Bahwašī, is an example of a genre ...

Much traditional scholarship holds that the period after about 1250 saw a decline in the production of scientific and philosophical works in the Arab world. This view is challenged by the impressive number of manuscripts written after that date in different Arabic-speaking countries that contain original treatises and commentaries. The work preserved in this manuscript, *Nuzhat al-Hussāb al-Muhtasara min al-Muršida* (The abridged amusement of the calculator from *The guide*), is a shorter version of *Muršida fī Sina’at al-Gubar *(The guide to the art of the numerals), an extensive treatise ...

The treatise preserved in this manuscript, *Al-Luma‘al-yasīra fī ‘ilm al-hisāb *(The little sparkles on the science of calculation), deals with Muslim inheritance. Of the social innovations that came with the Islamic conquest, the introduction of the system of *fara'id *(shares) for inheritances was one of the most radical and socially advanced. The fourth *surah* of the Qurʼan, verses 11–12, criticizes the traditional pre-Islamic system of agnatic succession, under which only men could inherit property, and provides for a proportional division among all the heirs, women included. The ...

The present manuscript preserves a very elegant copy of a work by one of the most prolific authors of the second half of the 15th century in the field of mathematics and related subjects: Badr al-Dīn Muḥammad ibn Muḥammad ibn al-Ġazal, best known as Sibṭ al-Māridīnī ("the son of al-Māridīnī’s daughter") from the name of his famous maternal ancestor, who was also a mathematician. The *Lum‘a al-Māridīnīyya *is an extensive prose commentary on a famous poem on algebra composed by the Maghrebi mathematician al-Yāsamīn around the last ...

The mathematical tradition that flourished in North Africa and Andalusia during the Middle Ages did not undergo the same decline that many scholars claim occurred in the sciences after the first half of the 13th century. The present work supports this point. The manuscript is a very elegant copy of a mathematical text by Badr al-Dīn Muḥammad ibn Muḥammad ibn al-Ġazal, best known as Sibṭ al-Māridīnī ("the son of al-Māridīnī’s daughter") from the name of his famous maternal grandfather, who was himself a mathematician. Sibṭ al-Māridīnī's mathematical ...

The 12th-century mathematical poem known as *al-Yāsamīnīyya fī ‘ilm al-Jabr *(The poem by al-Yāsamīn on calculus) from the name of its author, al-Yāsamīn, is one of the most read and commented upon mathematical texts of its time. Its verses have been extensively copied, both in autonomous form and by incorporation into larger commentaries up to the 20th century. The present manuscript preserves an early 20th-century copy of the 15th-century commentary on the *Yāsamīnīyya* written by Badr al-Dīn Muḥammad ibn Muḥammad ibn al-Ġazal, best known as Sibṭ al-Māridīnī ("the son ...

The system of *fara'i**ḍ *(shares) for inheritances is considered to be one of the most advanced innovations introduced by Muslim conquerors in Middle Eastern and North African societies. The exact calculation of shares of inheritance is a complex chapter in Islamic law, and it is not surprising that Muslim intellectuals and scientists developed a system of mathematical tools in order to master "the science of the shares" (*‘ilm al-fara'i**ḍ*). An important contribution to this field can be found in the work of Aḥmad ibn Muḥammad Ibn ...

The present manuscript preserves an extensive commentary on the 17th-century mathematical treatise *Al-**Ḫulāṣa fī al-Ḥisāb *(The abridgment on calculus), which was composed by Bahā' al-Dīn Al-‘Amilī (1547–1621), one of the leading intellectuals of 17th century Safavid Persia (present-day Iran). Born in the city of Baalbek (present-day Lebanon), Al-‘Amilī was an important figure in many different fields of knowledge, including theology, mysticism, poetry, astronomy, mathematics, and architecture. His main contribution to mathematics, the *Al-Ḫulāṣa fī al-Ḥisāb*, was well known and is the subject of the commentary by ...

This 18th-century manuscript offers a clear example of the continued use in the Islamic world of the scientific commentary well after the end of Middle Ages, the period most associated with Arabic scientific achievement and this literary form. In this case, the treatise commented upon is the *Nuzhat al-nuẓẓār fī ‘ilm al-ghubār *(The excursion of the observer in the science of numerals), which was itself an abridgment by Aḥmad ibn Muḥammad al-Farāḍī ibn al-Hā'im (around 1356-1412) of his own mathematical treatise entitled *Murshid al-ṭālib ilā asnā' al-maṭālib *(A student ...

The present manuscript preserves a copy of Aḥmad ibn Qāsim al-Shāfi‘ī al-Ġhazzī's *Kitāb sharḥ al-nuzha fī ‘ilm al-ḥisāb*(The book of the explanation of the excursion in the science of calculus), a work that exemplifies the lively interest in the mathematical sciences that persisted in the Islamic world well after the end of the "classical" period that saw the flowering of Arabic sciences. *Kitāb sharḥ al-nuzha fī ‘ilm al-ḥisāb*is technically a supercommentary. Al-Ġhazzī ’s work is an explanation of *Nuzhat al-nuẓẓār fī ‘ilm al-**ghubār *(The excursion ...

The Arabic mathematical tradition, which flourished during the Middle Ages, transmitted and enriched the knowledge derived from Greek and Indian sources. Arabic mathematicians further developed these studies, seeking to answer theoretical as well as practical problems. Medieval Arabic mathematical treatises were extensively copied, studied, and commented upon in subsequent centuries, as exemplified in this manuscript. This supercommentary (commentary on a commentary) by Aḥmad Muhammad al-Šāfiʻī al-Janājī al-Mālikī elucidates an earlier commentary by Zakarīyā ibn Muḥammad al-Anṣārī (circa 1420–1519) on a work by Aḥmad ibn Muḥammad al-Farāḍī ibn al-Hāʼim (circa ...

This supercommentary (commentary on a commentary) by Aḥmad Muhammad al-Šāfiʻī al-Janājī al-Mālikī testifies to the liveliness and endurance of the Arabic mathematical tradition and demonstrates the continuous exegetical effort in which Arabic scientists commented upon previous works with the aim of expanding and clarifying their contents. The North African mathematician ‘Abd Allāh ibn Ḥajjāj ibn al-Yāsamīn (died 1204) conveyed his mathematical knowledge in a poem known as *Yāsamīnīyya* (The treatise by al-Yāsamīn). Al-Yāsamīn’s verses became the subject of a prose commentary, the *Lum‘a al-Maridinīyya fī** Šarḥ al-Yāsamīnīyya *(The ...

This manuscript, probably dating from the 17th century, preserves only a section of what appears to have been an extensive and complete treatise on practical geometry. The title on the second page of the manuscript in fact states that it is “the third section of the book of the General Rules in the Science of Measurement.” The larger work of which this is a part consisted of four introductory essays, five chapters, and a conclusion. The author is unknown, as the opening of the treatise where indications of authorship might ...

This manuscript is an 18th-century copy of *Nuzhat al-ḥisāb* (The excursion of calculus)by mathematician Aḥmad ibn Muḥammad al-Farāḍī ibn al-Hā'im (around 1356–1412). The copyist of this brief but densely written codex provided his name and the date of the completion of his work in the colophon: at the bottom of the last page of the manuscript is stated that Aḥmad ibn Qāsim ibn Aḥmad ibn Muḥammad al-Hanbalī finished this copy in the year 1185 AH (1772). The *Nuzhat al-ḥisāb*is an abridgment by Ibn al-Hā'im of ...

Bahaa al-Din al-Amili (1547–1621 AD; 953–1031 AH) is thought to have been born in Baalbek, Lebanon, but his family moved to Herat, present-day Afghanistan, to escape Ottoman persecution. He studied in Isfahan, in present-day Iran, and continued on to Aleppo, Jerusalem, and Egypt, before returning to Isfahan, where he served for many years as the chief judge and where he died. He produced more than 50 titles in fields as diverse as arithmetic, astronomy, literature, religion, and linguistics. Known for his poetry as well as his many encyclopedic ...

Bahā’ al-Dīn Muḥammad ibn Ḥusayn al-‘Āmilī, also known as Sheiykh Bahā’ī, was a famous polymath and intellectual luminary of Safavid Persia. He was born in 1547 (953 AH) near the Jabal ‘Āmila in Syria. He migrated with his family to Persia (perhaps to escape the persecution of Shi'a Muslims at the hand of the Ottomans), where he eventually obtained an honored place at the court of Shāh ‘Abbās. He died in Isfahan in 1621 (1030 AH). A prolific author of works on astronomy, mathematics, *fiqh* (Islamic jurisprudence ...