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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 ...

In the middle of the 15th century, a number of manuscripts by the third-century BC Greek mathematician Archimedes began to circulate in the humanistic centers in the courts of Italy. Piero della Francesca (circa 1416–92), the Renaissance artist best known for the frescos he painted for the Vatican and for the chapels in Arezzo, transcribed a copy of a Latin translation of Archimedes’s geometry (a compilation of seven surviving treatises) and illustrated it with more than 200 drawings representing the mathematical theorems in the texts. This manuscript, long ...

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 Naṣīr al-Dīn Muḥammad ibn Muḥammad al-Ṭūsī (1201–74). After his death al-Ṭūsī was referred to as *al-muʿallim al-thālith* (the third teacher, with Aristotle and Fārābī referred to as the first and second teachers, respectively). An extraordinarily prolific author, al-Ṭūsī made notable contributions to most of the intellectual fields of his era, writing on theology, mysticism, logic ...

Johannes Huswirth (Sanensis) was a German arithmetician who flourished around 1500. Nothing is known of his life. That he is sometimes referred to as Sanensis suggests that he may have come from Sayn, Germany. *Arithmetice Lilium Triplicis Practice* (The threefold lily of practical arithmetic) presents basic arithmetic operations such as addition and multiplication for whole numbers and fractions. It treats much of the same material that Huswirth had covered in an earlier work, *Enchirdion Algorismi* (Handbook of algorithms). The work includes two woodcut illustrations; one of God the Father and ...

This work is a treatise on the construction and use of the weighing balance (*qabān*, also *qapān*). It brings together geometric, mechanical, and arithmetic knowledge needed to construct and utilize measuring devices for weighing heavy and irregularly-shaped objects. The author’s name is unknown, but excerpts from another work by an already-deceased Shaykh ‘Abd al-Majīd al-Shāmulī al-Maḥallī are quoted in the treatise. The last page of the manuscript contains a sheet of verses that describe the basics of using a weighing balance, in a form that is easy to remember ...

This work is a treatise on trigonometry by Li Madou, the Chinese name of the Italian Jesuit Matteo Ricci (1552–1610). Ricci left for China in 1581 and arrived in Macao in 1582. Together with Luo Mingjian (Michele Ruggieri, 1543–1607), he began his mission in Zhaoqing, Guangdong Province, where he published his *Wan guo yu tu* (Map of 10,000 countries), which was well received by Chinese scholars. He was expelled from Zhaoqing and went to Jiangxi, where in 1596 he became the superior of the mission. He lived ...

This mathematical treatise by Muḥammad b. Abi al-Fatḥ Muḥammad b. al-Sharafī Abi al-Rūḥ ‘Īsā b. Aḥmad al-Ṣūfī al-Shāfi‘ī al-Muqrī, was written in 1491-92 (897 AH). It begins with a "General Introduction," followed by two main parts, with a concluding section on the study of cubes and cube roots. Part I, "Operations on Simple Irrational Radicals," is divided into four chapters. Chapter 1 covers simplification of radicals. Chapters 2, 3, and 4 deal respectively with the multiplication, addition and subtraction, and division of radicals. Part II, on "Operations with Compound ...

This copy of *Mabādi' al-handasa *(Elements of geometry) is a second edition of a work by Rifā‘ah Rāfi‘ al-Ṭahṭāwī (1801−73), a leading intellectual and a pioneer of the 19th century Egyptian enlightenment. In his introduction, the author refers to an edition of 1842−43, written for students at the Madrasa al-Ṭubjīa, the military school founded by Muḥammad ʻAlī Bāshā (1769−1849) in Ṭura, Egypt. He also mentions the celebrated 1794 geometry textbook by A.M. Legendre, *Eléments de géométrie* (Elements of geometry). Al-Ṭahṭāwī says that this new 1854 ...

This manuscript is a didactic work on arithmetic and algebra, composed in versified form, as a *qasīda* of 59 verses. It was composed by Ibn al-Hā’im al-Fardī in 1402 (804 A.H.). The beginning of the work also names ‛Alī b. ‛Abd al-Samad al-Muqrī al-Mālikī (died Dhu al-Ḥijja 1381 [782 A.H.]), a scholar and teacher who had come to Egypt and taught at the ‛Amr b. ‛As madrasa for several years. The main part of the *qasīda* begins by introducing and defining key terms in arithmetic and algebra ...

This work is a tutorial text on elementary arithmetic, in 20 folios. It is divided into an introduction, 11 chapters, and a conclusion. In the beginning, the sign for zero is introduced, along with the nine Indian numerals, written in two alternative forms. This is followed by a presentation of the place system. The first four chapters cover, respectively, addition, subtraction, multiplication, and division. Chapter five introduces operations on non-whole numbers. The remaining six chapters discuss fractions and operations on them.

This work is a comprehensive tutorial guide on arithmetic and plane geometry, in 197 folio pages. It also discusses monetary conversion. The work is composed in verse form, and is meant as a commentary on existing textbooks. The author gives the following personal account of the writing of this guide: In Rajab 827 A.H. (May 1424) he traveled from Damascus to Quds al-Sharīf (in Palestine), where he met two scholars named Ismā‘īl ibn Sharaf and Zayn al-Dīn Māhir. There he took lessons on arithmetic, using an introductory book ...

This manuscript is a work on practical astronomy and the drawing of the circle of projection and related concepts from spherical trigonometry. It is rich with geometric diagrams, tables of empirical observations, and computations based upon these observations. An interesting feature of the manuscript is the appearance on the margins of the cover, and on several pages in the manuscript, of edifying verses, proverbs, and witty remarks. One reads, for example, “It is strange to find in the world a jaundiced physician, a dim-eyed ophthalmologist, and a blind astronomer.” Most ...

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 treatise by Badruddin al-Maridini (died 1506 [912 AH]),
better known as Sibt al-Maridini, includes an introduction, 20 sections, and a
conclusion. The treatise discusses a range of issues in astronomy, surveying,
and mathematics. It describes the sine quadrant and parallel circles, and
explains how to measure the width of a river, the angle of a star, the depth of
a well, or the height of a mountain. Al-Maridini, whose parents were from Damascus, was born, raised, and educated in Cairo late in the Mamluk
Dynasty (1250–1517). The manuscript ...

This work is a versified treatise on arithmetic (*‘ilam al- ḥisāb*), and specifically the art of dividing inheritance (*farā’iḍ*), which has application in Islamic law. After a standard expression of praise for the Prophet, his companions, and later followers, the text introduces the system of place values and explains multiplication of multi-digit whole numbers and simple and compound fractions. The text presents multiple examples that are described in verbal terms. As noted at the end of the manuscript, which was completed on Monday, 20 Rabī‘ I of the year ...

This work is a commentary on a versified, 59-line introduction to algebra, entitled *Al-Muqni‘ fī al-jabr wa al-muqābila*, by the prolific and influential mathematician, jurist, and man of letters Abū al-‘Abbās Shihāb al-Dīn Aḥmad ibn Muḥammad ibn ‘Alī al-Maqdisī al-Shāfi‘ī, known as Ibn al-Hā’im (circa 1356-1412 [circa 753-815 AH]). It clarifies the nomenclature and explains the basic concepts of algebra, and provides succinct examples. The manuscript, completed on Thursday night, 8 Sha‘bān 1305 AH (March 21, 1888), is in the hand of Tāhā ibn Yūsuf.

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 treatise on the art of arithmetic, completed in the late 1880s, opens a window into the early interaction between traditional and modern mathematical pedagogy in Egypt. The use of French loan words, such as *million*, along with some modern notation, indicates the author’s familiarity with developments in the teaching of arithmetic at the time. The work has an introduction followed by ten chapters and a conclusion. Following traditional praise for God, the Prophet Muhammad, and virtuous vanguards of learning, the treatise opens by introducing arithmetic as a useful ...

This guidebook is a short commentary on a work on arithmetic entitled *al-Wasīla *(The tool) completed in the 14th century by Shihāb al-Dīn Ahmad ibn Alī ibn Imād. The commentary is by the renowned Egyptian scholar known as Sibt (grandson of) al-Māridīnī (1423–1506), who taught mathematical sciences at Alazhar for a long time. The body of the work begins with a general discussion on numbers, and forms a standard introduction to arithmetic. The manuscript, which was completed by Ahmad ibn Yūnus al-Chalabī al-Hanafī in 1496 (AH 903) at the ...

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 ...

Arithmetics were a popular genre of textbooks during the era of the Bulgarian National Revival in the 19th century, when it was widely believed that everyone, especially future businessmen, needed to know basic mathematics. *Bulgarian Arithmetic* was the fourth such text published in this era, in 1845. The author, Khristodul Kostovich Sichan-Nikolov (1808–89), was a monk, teacher, writer, and publicist, often assisted in his scholarly pursuits by the writer, educator, and priest Neofit Rilski. Before writing his own text, Sichan-Nikolov had been involved as the editor of the first ...

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 Qur'an, 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 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 ...