Contributions of Muslim Scientists to Geometry

Sunday, 12 December 2010

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Written by Dr. Ragheb Elsergany

Introduction

Geometry was known by the ancient man to meet his need for measurement, whether spaces or construction. Some people may go farther and say that geometry is a primitive science; even animals realize that the straight line[1] is the shortest road between two points.

Geometry in Ancient Civilizations

Geometry dates back to ancient Egyptians as they applied the theory which was known later as Pythagorean Theorem. Their excellence is vivid clear in their monuments. There was a document, dating back to Amasis, who is also known as Ahmose, 4000 years ago; it contained geometrical thoughts about the areas and sizes of different shapes. Then, Babylonians introduced new additions, which were adopted by Greeks who excelled a lot in this field. Among the most famous Greek scientists was Euclid who authored the most famous book across history, entitled The Foundations of Geometry. This book was transferred to Europe through its Arab translation[2].

Geometry of Muslims

Geometry was introduced to Arabs and then Muslims through the translation of Greek works, especially the book of Euclid, Foundations of Geometry. Donald R. Hill[3] traced down the development of geometry in Islamic civilization, pointing out that "Following the phase of translation was the phase of innovation." Though masters like Euclid, Apollonius and Archimedes were duly appreciated, Arab scientists refuted their conclusions and corrected them in some cases. Besides, Arab scientists introduced unprecedented contributions to the field of theoretical geometry[4]. Astonishingly enough, these "unprecedented contributions" were centered on "theoretical geometry," a field which was not considered much by Muslims.

Types of Geometry for Muslim Scientists

Muslim scientists divided geometry into two types: mental and concrete. Mental geometry refers to theoretical geometry, and concrete geometry refers to practical applications. Muslim scientists did not add much to intellectual theoretical geometry; they merely explained and commented on it. They were mainly concerned with the concrete applied practical geometry. They applied it in the fields of industry, urbanity, arts, and construction[5] to the extent that the word "geometry" which was originally used to refer to theoretical geometry" has been used in Arabic to mean applied geometry[6].

Development of Geometry by Muslim Scientists

Geometry was markedly developed by Muslim scientists during the Islamic civilization. In some publications of Al-Biruni, there are some geometrical theories, givens and proofs. These methods were novel and profound, and differ from those adopted by Greek philosophers and mathematicians. Muslim scientists, including Ibn Al-Haytham, employed both plane and solid geometry to specify the reflection for statuses of spherical, cylindrical, conical, convex and concavo-convex, and they unprecedentedly introduced creative general solutions to them[7]. Muslim scientists pointed out that how to identify the proportion of the periphery of the circle to its diameter. They also proved excellent in plane geometry concerning parallels. Nasir Al-Din Al-Tusi[8] was the first one to draw the attention to prove that Euclid's theory lacks the issue of parallels; he introduced evidence based on hypotheses in his book Al-Resala Al-Shfia An Alshk Fil Khotot Al-Mtwazia (Adequate Treatise on Doubts about Parallel Lines). Muslim mathematicians know how the science of flattening the circle. Haji Khalifa viewed this science as the "science through which we know how the circle is transferred to a surface by keeping lines and circles drawn on the circle and how these drawn circles are transferred to a circle then to a line[9]." The importance of this science, according to Al-Qongy, lies in the fact that it can be used in other sciences, especially astronomy. Of their publications in this field of geometry are Al-Kamel by Farghani, Al-Isti'ab by Al-Biruni, and Dostour Al-Targih fi Kawaad Al- Tastih by –Taqi al-Din al-Shami[10], may Allah keep their souls in peace[11].

Muslim scientists introduced a lot of publications on geometric problems, geometric synthesis, angle divisions, drawing of regular rectangular shapes, and linking them to algebric equations. It is said that Thabit ibn Qurra[12] divided the angle into three equal sections using a way which was different from that known by Greeks.

Qadri Toqan pointed out that sines were used instead of hypotenuse at the beginning of the third hijjri century. It is said that it was Thabit ibn Qurra who introduced (Manalos claims), with their present form. Above all, he solved some of the cubic equations in geometrical ways sought by some western scientists in their mathematical research in the sixteenth century AD, including Cartan and other great mathematicians.

Qadri Toqan went on to say that those who are concerned with mathematics do not believe that Thabit was among those who paved the way for (calculus), a science which is of great significance for inventions and discoveries. But for all this science and its facilities for solving a lot of hard sums and operations, natural laws would not have been exploited for the benefit of humanity. Thabit was one of those concerned with analytic geometry and excelled in it. He introduced unprecedented innovations and wrote a book in which he explained the relation between algebra and geometry and how to combine them[13].

The French Orientalist Baron Carra de Vaux[14] described the achievements attained by Islamic civilization, pointing out that "Arabs introduced really great inventions; they taught us the usage of zero, although they did not invent it. Arabs also were the first to measure daily life and made algebra a perfect science and excelled in it. They, further, set the bases of analytical geometry. They were matchlessly the founders of plane triangles and spherical triangles sciences, which could not be ascribed to Greeks if accuracy and integrity were considered[15]. The great development in the field of science can be attributed to the Arabs' employment of Indian numbers, especially the zero whose discovery was controversial. However, it was undoubtedly used by Arabs, who considered it a representation of an empty digit or place. By using it, calculations which were based on digits or places could be made, and lengthy mathematical operations, which were impossible to calculate by using Latin numbers then could be made[16].

Sigrid Hunke, German Orientalist, pointed out that Arabs did not haphazardly or accidentally use these numbers; they could ingenuously conceive these numbers written above Indian gifts and goods. These Indian numbers had become that famous at the hands of Arabs[17].

"Mathematicians regard zero as the greatest human invention. Indeed, but for zero, positive and negative quantities in the science of electricity, and the positive and the negative in algebra could have never seen light[18]".

Another leap occurred in geometry when Al-Khwarizmi introduced algebra, which will be tackled when dealing with the contributions made by Muslims in human sciences.

In the field of area, the book, Marefet Mesahet Al-Ashkal Al-Basita Wa Al-Koreya (the Book of Measurement of the Plane and Spherical Figures) in geometry is considered one of the most important works of the Sons of Musa bin Shaker. In it, they emphasized the importance of identifying length, width and size.[19]

This book by the Sons of Musa bin Shaker constituted a significant development of the two books of Archimedes on Measurement of a Circle and on the Sphere and Cylinder. In it, they utilized an approach used by udox and the concept of meager quantities introduced by Archimedes. This book was of great importance for the Islamic East and the Latinized west alike[20].

Muslim scientists tackled areas in their mathematical publications as being a branch of geometry. For example, Bahauldin Al-Aamili[21] (1031 A.H./ 1622 A.D.) devoted the first three chapters of part six of his book Kholaset Al-Hesab( The Gist of Mathematics). In the introduction, he introduced basic definitions on area, especially the area of the surfaces and bodies. In chapter one, he dealt with the area of surfaces with straight sides as triangle, square, rectangle, rhombus, hexagon and octagon, among others. In chapters two and three, he focused on the way through which the area of circles, curved surfaces, such as cylinders, complete and incomplete cones and the circle. In part seven, he referred to some issues related to the area of the land surface to conduct surveying to dig canals and determine heights, width of rivers and depth of wells.

It was natural of Muslims to transfer their geometrical knowledge and apply it to their architectural art, depicted in masjids (mosques), places and cities, among others. They paid attention to geometrical decorations, which were characterized by symmetry and accuracy. Much research has been done on Islamic art, assuring the originality of architectural geometry, which has been reiterated by Martin S. Breks, an orientalist specialized in Islamic architecture. He viewed Islamic architecture as being original and distinct, pointing out that "even though Arabs were not probably aware of architectural geometry during the early period of conquests, it is vivid clear that Islamic architecture was similar in all countries and across the period under Islam. Though the roots of this architecture were extremely sophisticated (namely the sources of influence and copying), it has been distinct from the other local architectural schools where they were born[22]".

In view of the above, it is vivid clear that Muslims excelled in geometry; their role could not be ignored at all. Mohamed Kurd Ali pointed out that "As far as geometry is concerned, Muslims were peerless innovators. Arabs did not invent buildings of their own; their geometry was full of their love of decoration and nicety. They invented the propped arch and excelled in the geometry of domes, ceilings and suspended ceilings, made of trees and flowers for their mosques and palaces. All these decorations have rendered these places masterpieces. According to a foreign knowledgeable person, the excessive obsession of Muslims with decorations has made their buildings as an oriental dress, which was beautifully knit and ornamented[23]".

These have been some of Muslim contributions to the development of geometry. The characteristics of this science became clear after they had checked the heritage of the previous civilizations.

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