Muslim Contribution to Geometry

Muslim Contribution to Geometry

Dr. Ali Daffa’a

Mathematics had its origin during the earliest events in human history, from the time man found it necessary to count and to measure. These early activities stimulated the eventual development of independent subjects, arithmetic and geometry. Consequently, mathematics has a dual foundation and two main themes. Arithmetical procedures, counting and measurement, appear to have developed simultaneously with the passage of time. If either topic were eliminated, mathematics would be permanently damaged. “Any scheme of mathematical instruction which minimizes or ignores the indispensable role of either geometry or arithmetic seriously unbalances the curriculum, endangers the student’s progress, and leads inevitably to mathematical stagnation and inefficiency.”

Modem civilization has been focused on science and technology with modern science being a continuation of an ancient era, for modern civilization could not exist without scientific thought. For example, Euclid lived in Alexandria more than twenty – two centuries ago, yet his geometrical ideas are still very much alive. His name is often equated with that of geometry itself.

Euclid’s work on geometry entitled Book of Basic Principles and Pillars was the first Greek work to be translated for students in Muslim lands.

Translations of various works began under Al-Mansur and were further developed under his grandson, Al-Ma’mun. A prince with a fine intellect, a scholar, philosopher, and theologian, Al-Ma’mun was instrumental in the discovery and translation of the works of ancient people. During the reign of Harun Al-Rashid, Al-Hajjaj ibn Yusuf translated into Arabic several Greek works. Among these translations were the first six books of Euclid and the Almagest. The Almagest, written by Cludius Ptolemy of Alexandria, was the most outstanding ancient Greek work on astronomy.

The rationale for acquiring knowledge of geometry, as regarded by the Muslim mathematicians, is set forth in the writings of lbn Khaldun:

It should be known that geometry enlightens the intellect and sets one’s mind right. All its proofs are very clear -and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence. The following statement was written upon Plato’s door; “No one who is not a geometrician may enter our house.”

The work of the Muslims in the application of geometry to the solution of algebraic equations suggests they were the first to establish the close interrelation of algebra and geometry. This was a leading contribution toward the later development of analytic geometry,

The Muslims helped to advance mathematical thought during the Dark Ages. It was during the ninth and tenth centuries that they gave to Europe its first information about Euclid's Elements.

Definition of Geometry

Geometry is science which not only leads to the study of the properties of space, but also deals with the measurement of magnitude. It has as its objective the measurement of extension which has length, width, and height as its three dimensions. The word itself came originally from two Greek words, geo, meaning earth, and metria, measurement. It, therefore, meant the same as the word surveying, which is derived from the Old French, meaning “to measure the earth.”

The Muslims explained that the name of Euclid, which they called Uclides, was compounded by Ucli, meaning a key, and Dis or measure. When combined, they meant the ‘key of geometry." Euclid’s name has remained a synonym for geometry.

According to William David Reaves

Geometry came to be used to designate that part of mathematics dealing with pof nts, lines, surfaces, and solids or with some combination of these geometric magnitudes.

Origin of Geometry

The first geometrical considerations of mankind are ancient and seem to have their origin in simple observations, beginning from human ability to recognize physical iom1s by comparing shapes and sizes. There were innumerable circumstances in the life of primitive man that would lead to a certain amount of subconscious geometric discovery. Distance was one of the first geometrical concepts to be developed, and the estimation of the time needed to make a journey led to the belief that a straight line constituted the shortest path from one pof nt to another. It is apparent that even animals seem to realize this instinctively. The need to measure land led to the idea of simple geometric figures, such as, rectangles, squares, and triangles. When fencing a piece of land, the comers were marked first and then jof ned by straight lines. Other simple geometrical concepts, vertical, parallel, and perpendicular lines, would have originated through practical construction of walls and dwellings.

According to the Greek historian Herodotus (c. 450 B.C.), geometry originated in Egypt because the menstruation of land and the fixing of boundaries were necessitated by repeated inundations of the Nile. An ancient manuscript of the Egyptians, now in the British Museum in Lndon, and written by Ahmes, a scribe of about 2000 B.C., contains rules and formulas for finding areas of fields and capacities of wheat warehouses. During the period of its origin, about 1350 B.C., geometry was used largely as a means to measure plane figures and volumes of simple solids. The Egyptian, mathematicians excelled in the field of geometry and were in many respects, superior to the Babylonians. As a deductive science, geometry was started by Thales of Miletus (c. 600 B.C.), who introduced Egyptian geometry to Greece.

Ibn Al-Haitham

Aristotle and lbn Khaldun both considered optics as a branch of geometry. Progress made in the field of optics would certainly have been impossible in medieval times without the knowledge of Eclid’s Elements and Apollonius' conics. The science of optics explains the sons for errors in visual perception. Visual perception takes place through a cone formed by rays, in which the top is the pof nts of vision and the base is the object seen. Close objects appear large and distant images appear small. Furthermore, small objects appear large under water or behind transparent bodies. Optics seeks to explain these scientific phenomena with geometrical proofs. Optics also presents an explanation of the differences in the perspective view of the moon at various latitudes. Knowledge of the visibility of the new moon and of the occurrence of eclipses is based on these conjecture.

A great stimulus to optical investigation was provided in the first half of the eleventh century by lbn al-Haitham (Alhazen). The Muslim mathematician was the first scholar to attempt to refute the optical doctrines of Euclid and Ptolemy. According to those doctrines, the eye received images of various objects by sending visual rays to certain pof nts. In his book on optics, Al-Haitham proved that the process is actually the reverse and thus laid the foundations of modem optics. His formula was that it is not a ray that leaves the eye and meets the objects that gives rise to vision, but rather that the form of the perceived object passes into the eye and is transmitted by the lens.

Geometry was used extensively by Al-Haitham in his study of optics. His work on optics, which included the earliest scientific account of atmospheric retraction contained a geometrical solution to the problem of finding a pof nt on a concave mirror; that a ray from a given pof nt must be incident in order to be reflected to another given pof nt. Al-Haitham also discovered some original geometrical theorems such as the theorem of the radical axis.

The works of lbn al-Haitham became known in Europe during the twelfth and thirteenth centuries. Joseph ibn .'Aqnin referred to lbn al-Haitham’s work in optics as being greater than those of Euclid and Ptolemy. Al-Haitham’s optics were made known to European mathematicians at about the same time by John Peckham, the Archbishop of Canterbury, in 1279, and by the Polish physicist, Witelo.

Al-Haitham established the fundamental basis which eventually led to the discovery of magnifying lenses in Italy. Most of the medieval writers in the field of optics, including Roger Bacon, used his findings as their beginning. They particularly used Opticae Thesaurus, Al-Haitham’s book which was very important to Leonardo da Vinci and Johann Kepler. During the seventeeth century Al-Haitham’s work was very useful to the famous Kepler. The writings of Al-Haitham are “rooted in very sound mathematical knowledge, a knowledge that enabled him to propound  revolutionary doctrines on such subjects as the halo and the rainbow, eclipses and shadows, and on spherical and parabolic mirrors.”

Prior to his death in Cairo, Al-Haitham issued a collection of problems similar to the Date of Euclid. He, is known to have written nearly two hundred works on mathematics, physics, astronomy, and medicine. He also wrote commentaries on Aristotle and the Roman physician, Galen. Although he made major contributions to the field of mathematics, it is especially in the realm of physics that he made his outstanding achievements. He was an accurate observer and experimenter, as well as a theoretician.

Howard Eves has observed:

The name Al-Haitham  (965-1039), has been preserved in mathematics in connection with the so-called problem of Alhazen: To draw from two given pof nts in the plane of a given circle lines which intersect on the circle and make equal angles with the circle at that pof nt. The problem leads to a quartic equation which was solved in Greek fashion by an intersecting hyperbola and circle.  Alhazen was born in Basra in South Iraq and was perhaps the greatest of the Muslim physicists. The above problem arose in connection with his optics, a treatise that later had great influence in Europe.

The following is a partial list of Al-Haitham's works on geometry as appears in the Thirteen Books of Euclid’s Elements, Vol. I:

1.     Commentary and abridgment of Elements.

2.     Collection of the Elements of Geometry and Arithmetic, drawn from the treatises of Euclid and Apollonius.

3.     Collection of the Elements of the Calculus deduced from the principles laid down by Euclid in his Elements.

4.     Treatise on ‘measure’ alter the manner of Euclid's Elements.

5.     Memoir on the solution of the difficulties in Book I.

6.     Memoir for the solution of a doubt about Euclid, relative to Book V.

7.     Memoir on the solution of a doubt about the stereometric portion.

1.     B. Memoir on the solution of a doubt about Book XII.

8.     Memoir on the division of the two magnitudes mentioned in Book X. (Theorem of exhaustion).

9.     Commentary on the definitions in the work of Euclid.

Ibn at-Haitham tried to prove Euclid's fifth postulate. The Greeks’ attempt to prove the postulate had become a “fourth famous problem of geometry,” and several Muslim mathematicians continued the effort. Al-Haitham started his proof with a tri-rectangular quadrilateral (sometimes known as “Lambert‘s quadrangle” in recognition of his efforts in the eighteenth century). lbn al-Haitham thought that he had proved the fourth angle must always be a right angle. From this theorem on the quadrilateral, the fifth postulate is shown to follow. in his “proof” he assumed that the locus of a point that remains equidistant from a given line is necessarily a line parallel to the given line, which is an assumption shown in modem times to be equivalent to Euclid's postulate.

According to Hakim Muhammad Said, president of Hamdard National Foundation, Karachi:

In this year of grace, when man has first set foot on the moon and is reaching out to other stars, it is salutary to remember and acknowledge the great debt that modern mathematics and technology owe to the patient and exacting work of the early pioneers. This year we celebrate the 1,000 anniversary of one of the greatest of them, Abu Ali al-Hasan ibnul Hasan ibn ai-Haitham... ibn al-Haitham was a man of many parts, mathematician, astronomer, physicist, and physician. He had a 20th century mind in a 10^th century setting and his contributions to knowledge were quite extraordinary.

Thabit ibn Kurra

Thabit ibn Kurra (836-911 A.D.) of Harran, Mesopotamia, is often regarded as the greatest Muslim geometer. He carried on the work of Al-Khwarizmi and translated into Arabic seven of the eight books of the conic sections of Apollonius. He also translated certain works of Euclid, Archimedes, and Ptolemy which became standard texts.

Archimedes’ original work on the regular heptagon has been lost, but the Arabic translation by Thabit ibn Kurra proves the Greek still exists. Carl Schoy found the Arabian manuscript in Cairo, and revealed it to the Western public. It was translated into German in 1929.

Ibn Kurra wrote several books on the subject of geometry. A partial list of his works include: On the Premises (Axioms, Postulates, etc.) of Euclid. On the Propositions of Euclid, and a book on the propositions and questions which arise when two straight lines are cut by a third (the “proof” of Euciid’s famous postulate). He is also credited with Introduction to the Book of Euclid, which is a treatise on geometry.

The starting point for all geometric study among Muslims was Euclid’s Elements. lbn Kurra developed new propositions and studied irrational numbers. He also estimated the distance to the sun and computed the length of the solar year. He solved a special case of the cubic equation by the geometric method, to which lbn Haitham had given particular attention in 1000 A.D. This was the solution of cubic equations of the form X³ + a²b = cx² by finding the intersection of cx² = ay (a parabola) and y(c - x) = ab (a hyperbola). .

Other Muslim Geometers

Al-Kindi, who made significant contributions in the field of arithmetic, also worked in the area of geometry. His most important contributions to scientific knowledge was his work on optics, dealing with reflection of light, and his treaties on the concentric structure of the universe. Using a geometrical model, Al-Kindi gave a proof of the following:

1. The body of the universe is necessarily spherical .

2. The earth will necessarily be spherical and (located) at the center of the universe.

3. It is not possible that the surface of the water be non-spherical.

Al-Kindi wrote many works on spherical geometry and its application to the universe. The following is a partial list of his works on spherics:

1. Manuscript on “The body of the universe is necessarily spherical.”

2. Manuscript on “The Simple Elements and the Outermost Body are Spherical in Shape.”

3. Manuscript on “Spherics.”

4. Manuscript on “The construction of an Azimuth on a Sphere.”

5. Manuscript on “The surface on the water of the sea is spherical.”

6. Manuscript on “How to level a sphere.”

7. Manuscript on “The form of a skeleton sphere representing the relative positions of the Ecliptic and other Celestial circles.”

According to Florien Cajori the algebra of Al-Khwarizmi contained some geometrical ideas. He not only gave the theorem of the right triangle when the right triangle is isosceles, but also calculated the areas of the triangle, parallogram, and circle. For he used the approximation 3-1/7. One chapter in Al-Khwarizmi's algebra on mensuration dealt only with geometry and is called Bab al-Misaha. if Al-Khwarizmi had really studied Greek mathematics, there would certainly have been some traces of the contents or terminology of Euclid's Elements in his geometry. There are none. “Euclid's Elements in their spirit and letter are entirely unknown to him.”

Al-Hajjaj ibn Yusuf, Muslim geometer, translated the Elements of Euclid for Harun al-Rashid (786-809 A.D.) renaming the work Haruni. Al-Hajjaj revised his first translation for Al-Ma’mun (813-833 A.C.), the Caliphate, and the revised work was known as Al-Ma’mun.

The translation of the Elements of Euclid by Al-Haijaj did not include Book X, which was later translated with Pappus’ commentary by Sa'id A-Dimishqi.

Summary

The Muslims emphasized the study of geometry in their curriculum because it possessed practical applications in surveying, astronomy, and it aided the study of algebra and physics. Muslim geometry could be divided into constructional and arithmetical branches. When constructions were involved, the Muslims expressed the elements of geometrical figures in terms of of one another, that is, by the Methods of Greek geometry. Al-Khwarizmi was representative of this approach, with the solutions involving no arithmetical or algebraic technique. However, the numerical approach was more characteristic of Muslim geometry.

According to Suter:

“In the application of arithmetic and algebra to geometry, and conversely in the solutions of algebraic problems by geometric means, the Muslims for surpassed the Greek and Hindus.

The work of ibn al-Hailham on optics was the outstanding Muslim work in the area of applied geometry. On his work, while using geometry most effectively, he also contributed to the development of the subject with his work on the radical axis. Thabit ibn Kurra’s translation of Archimedes’ work on the regular heptagon saved the manuscript from being lost forever. Ibn Kurra also contributed several original texts based on the work of Euclid, and he generalized the Pythagorean theorem.

Finally, as the signs of mathematical awakening of Europe appeared in the thirteenth century, the Greek classics were available for translation. As the Christian monks made contact with Muslim universities in Spain, opening the way to the Renaissance, Euclid's Elements were translated again, but this time from Arabic to Latin.