Jamshīd al-Kāshī’s Miftāḥ al-Ḥisab (Key to Arithmetic) was largely unknown to researchers until the mid-20th century, and has not been translated to English until now. This is the second book in a multi-volume set that finally brings al-Kāshī’s groundbreaking textbook to English audiences in its entirety. As soon as it was studied by modern researchers, it changed some false assumptions about the history of certain topics in mathematics. Written as a textbook for students of mathematics, astronomy, accounting, engineering, and architecture, Miftah covers a wide range of topics in arithmetic, geometry, and algebra. By sharing al-Kāshī’s most comprehensive work with a wider audience, this book will help establish a more complete history of mathematics, and extend al-Kāshī’s influence into the 21st century and beyond. The book opens by briefly recounting al-Kāshī’s biography, so as to situate readers in the work’s rich historical context. His impressive status in the kingdom of Ulugh Beg is detailed, as well as his contributions to both mathematics and astronomy. As a master calculator and astronomer, al-Kāshī’s calculations of 2π and sin(10) were by far the most accurate for almost two centuries. His law of cosines is still studied in schools today. This translation contributes to the understanding and appreciation of al-Kāshī’s esteemed place in the scientific world. A side-by-side presentation of the source manuscript—one of the oldest known copies—and the English translation is provided on each page. Detailed footnotes are also provided throughout, which will offer readers an even deeper look at the text’s mathematical and historical basis. Researchers and students of the history of mathematics will find this volume indispensable in filling in a frequently overlooked time period and region. This volume will also provide anybody interested in the history of Islamic culture with an insightful look at one of the mathematical world’s most neglected figures.

Jamshīd al-Kāshī’s Miftāḥ al-Ḥisab (Key to Arithmetic) was largely unknown to researchers until the mid-20th century, and has not been translated to English until now. This is the second book in a multi-volume set that finally brings al-Kāshī’s groundbreaking textbook to English audiences in its entirety. As soon as it was studied by modern researchers, it changed some false assumptions about the history of certain topics in mathematics. Written as a textbook for students of mathematics, astronomy, accounting, engineering, and architecture, Miftah covers a wide range of topics in arithmetic, geometry, and algebra. By sharing al-Kāshī’s most comprehensive work with a wider audience, this book will help establish a more complete history of mathematics, and extend al-Kāshī’s influence into the 21st century and beyond. The book opens by briefly recounting al-Kāshī’s biography, so as to situate readers in the work’s rich historical context. His impressive status in the kingdom of Ulugh Beg is detailed, as well as his contributions to both mathematics and astronomy. As a master calculator and astronomer, al-Kāshī’s calculations of 2π and sin(10) were by far the most accurate for almost two centuries. His law of cosines is still studied in schools today. This translation contributes to the understanding and appreciation of al-Kāshī’s esteemed place in the scientific world. A side-by-side presentation of the source manuscript—one of the oldest known copies—and the English translation is provided on each page. Detailed footnotes are also provided throughout, which will offer readers an even deeper look at the text’s mathematical and historical basis. Researchers and students of the history of mathematics will find this volume indispensable in filling in a frequently overlooked time period and region. This volume will also provide anybody interested in the history of Islamic culture with an insightful look at one of the mathematical world’s most neglected figures.

Jamshīd al-Kāshī’s Miftāḥ al-Ḥisab (Key to Arithmetic) was largely unknown to researchers until the mid-20th century, and has not been translated to English until now. This is the third and final book in a multi-volume set that finally brings al-Kāshī’s groundbreaking textbook to English audiences in its entirety. As soon as it was studied by modern researchers, Miftah changed some false assumptions about the history of certain topics in mathematics. Written as a textbook for students of mathematics, astronomy, accounting, engineering, and architecture, Miftah covers a wide range of topics in arithmetic, geometry, and algebra. By sharing al-Kāshī’s most comprehensive work with a wider audience, this book will help establish a more complete history of mathematics, and extend al-Kāshī’s influence into the 21st century and beyond. The book opens by briefly recounting al-Kāshī’s biography, so as to situate readers in the work’s rich historical context. His impressive status in the kingdom of Ulugh Beg is detailed, as well as his contributions to both mathematics and astronomy. As a master calculator and astronomer, al-Kāshī’s calculations of 2π and sin(10) were by far the most accurate for almost two centuries. His law of cosines is still studied in schools today. This translation contributes to the understanding and appreciation of al-Kāshī’s esteemed place in the scientific world. A side-by-side presentation of the source manuscript—one of the oldest known copies—and the English translation is provided on each page. Detailed footnotes are also provided throughout, which will offer readers an even deeper look at the text’s mathematical and historical basis. Researchers and students of the history of mathematics will find this volume indispensable in filling in a frequently overlooked time period and region. This volume will also provide anybody interested in the history of Islamic culture with an insightful look at one of the mathematical world’s most neglected figures.

The Encyclopaedia fills a gap in both the history of science and in cultural stud ies. Reference works on other cultures tend either to omit science completely or pay little attention to it, and those on the history of science almost always start with the Greeks, with perhaps a mention of the Islamic world as a trans lator of Greek scientific works. The purpose of the Encyclopaedia is to bring together knowledge of many disparate fields in one place and to legitimize the study of other cultures' science. Our aim is not to claim the superiority of other cultures, but to engage in a mutual exchange of ideas. The Western aca demic divisions of science, technology, and medicine have been united in the Encyclopaedia because in ancient cultures these disciplines were connected. This work contributes to redressing the balance in the number of reference works devoted to the study of Western science, and encourages awareness of cultural diversity. The Encyclopaedia is the first compilation of this sort, and it is testimony both to the earlier Eurocentric view of academia as well as to the widened vision of today. There is nothing that crosses disciplinary and geographic boundaries, dealing with both scientific and philosophical issues, to the extent that this work does. xi PERSONAL NOTE FROM THE EDITOR Many years ago I taught African history at a secondary school in Central Africa.

This volulme features eight original papers dedicated to the theme “Persian Architecture and Mathematics,” guest edited by Reza Sarhangi. All papers were approved through a rigorous process of blind peer review and edited by an interdisciplinary scientific editorial committee. Topics range from symmetry in ancient Persian architecture to the elaborate geometric patterns and complex three-dimensional structures of standing monuments of historical periods, from the expression of mathematical ideas to architectonic structures, and from decorative ornament to the representation of modern group theory and quasi-crystalline patterns. The articles discuss unique monuments Persia, including domed structures and two-dimensional patterns, which have received significant scholarly attention in recent years. This book is a unique contribution to studies of Persian architecture in relation to mathematics.

Contemporary technical architectural drawings, in establishing a direct relationship between the drawing and its object, tend to privilege the visible physical world at the expense of the invisible intangible ideas and concepts, including that of the designer’s imagination. As a result, drawing may become a utilitarian tool for documentation, devoid of any meaningful value in terms of a kind of knowledge that could potentially link the visible and invisible. This book argues that design drawings should be recognized as intermediaries, mediating between the world of ideas and the world of things, spanning the intangible and tangible. The notion of the 'Imaginal' as an intermediary between the invisible and visible is discussed, showing how architectural drawings lend themselves to this notion by performing as creative agents contributing not only to the physical world but also penetrating the realm of concepts. The book argues that this 'in-between' quality to architectural drawing is essential and that it is critical to perceive drawings as subtle bodies that hold physical attributes (for example, form, proportion, color), highly evocative, yet with no matter. Focusing on Islamic geometric architectural drawings, both historical and contemporary, it draws on key philosophical and conceptual notions of imagination from the Islamic tradition as these relate to the creative act. In doing so, this book not only makes important insights into the design process and act of architectural representation, but more broadly it adds to debates on philosophies of the imagination, linking both Western and Islamic traditions.

A History of Mathematics: From Mesopotamia to Modernity covers the evolution of mathematics through time and across the major Eastern and Western civilizations. It begins in Babylon, then describes the trials and tribulations of the Greek mathematicians. The important, and often neglected, influence of both Chinese and Islamic mathematics is covered in detail, placing the description of early Western mathematics in a global context. The book concludes with modern mathematics, covering recent developments such as the advent of the computer, chaos theory, topology, mathematical physics, and the solution of Fermat's Last Theorem. Containing more than 100 illustrations and figures, this text, aimed at advanced undergraduates and postgraduates, addresses the methods and challenges associated with studying the history of mathematics. The reader is introduced to the leading figures in the history of mathematics (including Archimedes, Ptolemy, Qin Jiushao, al-Kashi, al-Khwarizmi, Galileo, Newton, Leibniz, Helmholtz, Hilbert, Alan Turing, and Andrew Wiles) and their fields. An extensive bibliography with cross-references to key texts will provide invaluable resource to students and exercises (with solutions) will stretch the more advanced reader.