The Muslim Contribution to Geometry

by Ali Abdullah al-Daffa

This article was extracted from Ali Abdullah Al-Daffa’s book The Muslim Contribution to Mathematics, London: Croom Helm, 1978, pp. 81-92.

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 had a dual foundation and two main themes. Arithmetical procedures, counting and measurement, appear to have developed simultaneously with the passage of time. 1

Modern civilization is positively based on science and technology; modern science being a continuation of an ancient endeavour. Modern civilization could not exist without scientific thought. 2 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. 3

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 name ‘Almagest’ is a Latinized version of the Arabic title Almagesti). 4 The Almagest, written by Claudius Ptolemy of Alexandria, was the most outstanding ancient Greek work on astronomy. 5

The rationale for acquiring a knowledge of geometry, as regarded by the Muslim mathematicians, is set forth in the writings of Ibn 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.” 6

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. 7 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. 8

Definition of Geometry

Geometry is a science 9 which not only leads to the study of the properties of space, 10 but also deals with the measurement of magnitude. 11 It has as its objective the measurement of extension which has length, width, and height as its three dimensions. 12 The word itself came originally from two Greek words, geo meaning earth, and metre, measurement. It, therefore, meant the same as the word surveying, 13 which is derived from the Old French, meaning ‘to measure the earth.’ 14

The Muslims used an interesting folk-etymology to explain the name of Euclid which was used in form Uclides; this word was thought to the composed of Ucli, meaning a key, and Dis, meaning a measure. When combined, the name meant the ‘key of geometry.’ 15 Euclid’s name has since then remained a synonym for geometry. 16 According to William David Reeve:

Geometry came to be used to designate that part of mathematics dealing with points, lines, surfaces, and solids or with some combination of these geometric magnitudes. 17

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 forms by comparing shapes and sizes. 18 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 point 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 corners were marked first and then joined by straight lines. Other simple geometrical concepts, vertical, parallel, and perpendicular lines, would have originated through practical construction of walls and dwellings. 19

According to the Greek historian Herodotus (c. 450 BC), 20 geometry originated in Egypt because the mensuration of land and the fixing of boundaries were necessitated by repeated inundations of the Nile. 21 An ancient manuscript of the Egyptians named ‘Papyrus Rhind,’ now in the British Museum in London, and written by Ahmes, a scribe of about 2000 BC, contains rules and formulas for finding areas of fields and capacities of wheat warehouses. 22 During the period of its origin, about 1350 BC, geometry was used largely as a means to measure plane figures and volumes of simple solids. 23 The Egyptian mathematicians excelled in the field of geometry and were equal to the Babylonians. 24 As a deductive science, geometry was started by Thales of Miletas (c. 600 BC), 25 who also introduced Egyptian geometry to Greece. 26

Ibn Al-Haitham

Iraqi 10,000 Dinar banknote featuring Ibn Al-Haitham

Iraqi 10,000 Dinar banknote featuring Ibn Al-Haitham

Aristotle and Ibn 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 Euclid’s Elements and Apollonius’ Conies. 27 The science of optics explains the reasons for errors in visual perception. Visual perception takes place through a cone formed by rays, in which the top is the point of vision and the base is the object seen. All objects appear larger if they are close to and smaller if they are distant from the observer. Furthermore, objects appear larger under water or behind transparent bodies. 28 Optics seek to explain these scientific phenomena by geometric means. Optics also presents an explanation of the differences in the perspective view of the moon at various latitudes. Knowledge of the phases of the moon and of the occurrence of eclipses is based on these conjectures. 29

A great stimulus to optical investigation was provided in the first half of the eleventh century by Ibn Al-Haitham (Alhazen). 30 This Muslim mathematician was the first scholar to attempt to refute the optical doctrines to Euclid and Ptolemy. According to these doctrines, the eye received images of various objects by sending visual rays to certain points. In his book in optics, Al-Haitham proved that the process is actually the reverse and thus laid the foundations of modern optics. His formula was that it is not a ray that leaves the eye and meets the object that gives rise to vision, but rather that the form of the perceived object passes into the eye and is transmitted by the lens. 31

Geometry was used extensively by Al-Haitham in his study of optics. His work on optics, which included one of the earliest scientific accounts of atmospheric refraction, contained a geometrical solution to the problem of finding the focal point of a concave mirror; that a ray from a given point must be incident in order to be reflected to another given point. 32 Al-Haitham also discovered some original geometrical theorems such as the theorem of the radical axis. 33

The works of Ibn Al-Haitham became known in Europe during the twelfth and thirteenth centuries. Joseph ibn ‘Aqnin referred to Ibn Al-Haitham’s work in optics as being greater than those of Euclid and Ptolemy. 34 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. 35

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 (Compendium of Optics), Al-Haitham’s book which was very important to Leonardo da Vinci and Johann Kepler. 36 During the seventeenth century Al-Haitham’s work was very useful to the famous Kepler. 37 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.’ 38

Prior to his death in Cairo, Al-Haitham issued a collection of problems similar to the Data of Euclid. 39 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 contributions. He was an accurate observer and experimenter, as well as a theoretician. 40 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 points in the plane of a given circle lines which intersect on the circle and make equal angles with the circle at that point. 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. 41

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

  • 1 Commentary and abridgement of the Elements
  • 1 Collection of the Elements of Geometry and Arithmetic, drawn from the treatises of Euclid and Apollonius
  • 2 Collection of the Elements of the Calculus deduced from the principles laid down by Euclid in his Elements
  • 2 Treatise on ‘measure’ after the manner of Euclid’s Elements
  • 3 Memoir on the solution of difficulties in Book I
  • 3 Memoir for the solution of a doubt about Euclid, relative to Book V
  • 4 Memoir on the solution of a doubt about the stereometric portion
  • 4 Memoir on the solution of a doubt about Book XII
  • 5 Memoir on the division of the two magnitudes mentioned in Book X (Theorem of exhaustion)
  • 6 Commentary on the definitions in the work of Euclid. 42

Ibn Al-Haitham tried to prove Euclid’s fifth postulate. The Greek’s 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 trirectangular quadrilateral (sometimes known as ‘Lambert’d quadrangle’ in recognition of Lambert’s efforts in the eighteenth century). Ibn 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 modern times to be equivalent to Euclid’s postulate. 43

According to Hakim Mohammed Said, President of Hamdard National Foundation in 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 ibn Al-Hasan ibn Al-Haitham .. . Ibn Al-Haitham was a man of many parts, mathematician, astronomer, physicist, and physician. He had a 20th century mind in a 10th century setting and his contributions to knowledge were quite extraordinary. 44

Thabit Ibn Qurra

Thabit ibn Qurra (836-911 AD) of Harran, Mesopotamia, is often regarded as the greatest Arab geometer. 45 He carried on the work of Al-Khwarizmi and translated into Arabic seven of the eight books of the conic sections of Apollonius. 46 He also translated certain works of Euclid, Archimedes, and Ptolemy which became standard texts. 47

Archimedes’ original work on the regular heptagon has been lost, but the Arabic translation by Thabit ibn Qurra proves the Greek manuscript still existed at the time of translation. Carl Schoy found the Arabian manuscript in Cairo, and revealed it to the Western public. It was translated into German in 1929. 48

Ibn Qurra wrote several books on the subject of geometry. A partial list of his works includes: 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 Euclid’s famous postulate). He is also credited with Introduction to the Book of Euclid, which is a treatise on geometry. 49

The starting point for all geometric studies among Muslims was Euclid’s Elements. 50 Ibn Qurra developed new propositions and studied irrational numbers. He also estimated the distance to the sun and computed the length of the solar year. 51 He solved a special case of the cubic equation by the geometric method, to which Ibn Haitham had given particular attention in 1000 AD. This was the solution of cubic equations of the form x3 + a3b = cx2 by finding the intersection of x2 = ay (a parabola) and y(c – x) = ab (a hyperbola). 52

Other Muslim Geometers

Al-Kindī

Al-Kindī, who made significant contributions in the field of arithmetic, also worked in the area of geometry. His most important contribution to scientific knowledge was his work on optics, dealing with the reflection of light, and his treatise on the concentric structure of the universe. 53Using a geometrical model, Al-Kindī 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. 54
Egyptian stamp featuring Al-Kindi

Egyptian stamp featuring Al-Kindi

Al-Kindī 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’ 55
  7. Manuscript on ‘The form of a skeleton sphere representing the relative positions of the ecliptic and other celestial circles’. 56

Al-Khwarizmi

Al-Khwarizmi’s algebra also contained some geometrical ideas, according to Florian Cajori. He not only gave the theorem of the right triangle when the right triangle is isosceles, but also calculated the areas of the triangle, parallelogram, and circle. For he used the approximation 3. 57 One chapter in Al-Khwarizmi’s Algebra on mensuration dealt only with geometry and is called Bab al-Misaha (Chapter on measurement of areas). 58 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. 59 Euclid’s Elements in their spirit and letter are entirely unknown to him. 60

Stamp featuring Al-Khwarizmi

Stamp featuring Al-Khwarizmi

Al-Hajjaj ibn Yusuf

Al-Hajjaj ibn Yusuf, Muslim geometer, translated the Elements of Euclid for Harun al-Rashid (786-809 AD), renaming the work ‘Haruni.’ Al-Hajjaj revised his first translation for Al-Ma’mun (813-33 AD), the Caliph, and the revised work was known as Al-Ma’mun. 61

The translation of the Elements of Euclid by Al-Hajjaj did not include Book X, which was later translated with Pappus’ commentary by Sa’id ad-Dimishqi. 62

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 can be divided into constructional and arithmetical branches. When constructions were involved, the Muslims expressed the elements of geometrical figures in terms 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 far surpassed the Greeks and Hindus.” 63

The work of Ibn Al-Haitham on optics was the outstanding Muslim work in the area of applied geometry. In his work, Al-Haitham challenged the doctrines of Euclid and Ptolemy. While using geometry most effectively, he also contributed to the development of the subject with his work on the radical axis. Thabit ibn Qurra’s translation of Archimedes’ work on the regular heptagon saved the manuscript from being lost forever. Ibn Qurra 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.

Notes:

  1. Maic Berge, Risala Abi Hayyan Fi l-‘Ulm: D’ Abu Hayyan al-Tawhidi (Paris, Extrait du Bulletin d’Etudes Orientates de L’Institut Francais De Damas Tome, XVIII, 1963-4), p. 289.
  2. George Sarton, Ancient Science and Modem Civilization (Loncoln, Nebraska, University of Nebraska, 1954), p. 8.
  3. University Library, Cambridge, England, Arabic MSS, 1075, fol. (00. 6.55).
  4. Sir Thomas Arnold and Alfred Guillaume, Legacy of Islam (London, Oxford University Press, 1949), p. 380.
  5. Howard Eves, An Introduction to the History of Mathematics (New York, Holt, Rinehart and Winston, 1969), p. 90.
  6. Abd-ar-Rahman ibn Muhammad ibn Khaldun al-Hadrami, The Muguaddemah’s ibn Khaldun (New York, Bollingen Foundation, 1958), pp. 130-1.
  7. Shibli, Recent Developments in the Teaching of Geometry (York, Pennsylvania, The Maple Press Company, 1932), p. 16.
  8. William David Reeve, Mathematics for the Secondary School (New York, Henry Holt and Company, 1954), p. 373.
  9. Olinthus Gregory, Mathematics for Practical Men (Philadelphia, T. K. and P. G. Collins, 1838), p. 104.
  10. James McMahon, Elementary Plane Geometry (New York, American Book Company, 1903), p. 1.
  11. Edward Rutledge Robbins, Plane Geometry (New York, American Book Company, 1906), p. 11.
  12. Charles S. Venable, Elements of Geometry (New York, University Publishing Company, 1875), p. 19.
  13. H. A. Freebury, A History of Mathematics: For Secondary Schools (London, Cassell and Company, 1958), p. 32.
  14. Hay ward R. Alker, Jr., Mathematics and Politics (New York, The Macmillan Company, 1968), pp. 1-2.
  15. Sir Thomas Heath, A History of Greek Mathematics (London, Oxford University Press, 1921), Vol. I, p. 355.
  16. George Sarton, Introduction to the History of Science (Baltimore, The Williams and Wilkins Company, 1953), Vol. II, Part I, p. 9.
  17. William David Reeve, ‘The Teaching of Geometry,’ The National Council of Teachers of Mathematics, Fifth Yearbook (New York, Teachers College, Columbia University, 1930), p. 1.
  18. Howard Eves, A Survey of Geometry (Boston, Allyn and Bacon, 1963), Vol. I, p. 1.
  19. M. A. Craig, Al-Handasah Attahliliyah (Cairo, Mutba’att Al-Ma’arif we Maktabatiha bi Masr, 1928), Vol. I, pp. 5-6.
  20. David Eugene Smith, History of Mathematics (New York, Dover Publications, 1958), Vol. I, p. 81.
  21. D. M. Y. Somerville, The Elements of Non-Euclidean Geometry (New York, Dover Publications, 1958), p. 1.
  22. James B. Dodd, Arithmetic (New York, Pratt, Oakley and Company, 1857),p. 1.
  23. A. Wilson Goodwing and Glen D. Vannatta, Geometry (Columbus, Ohio, Charles E. Merreill Books, 1961), p. 1.
  24. Solomon Gandz, ‘A Few Notes on Egyptian and Babylonian Mathematics,’ Studies and Essays in the History of Science and Learning (Offered in Homage to George Sarton on the occasion of his sixtieth birthday) (New York, Henry Schumann, 1944), p. 460.
  25. The Role of Mathematics in Civilization,’ The Place of Mathematics in Secondary Education, Fifteenth Yearbook of the National Council ¶of Teachers of Mathematics (New York, Bureau of Publications of Teachers College, Columbia University, 1940), p. 3.
  26. Benjamin Farrington, Science in Antiquity (London, Oxford University Press, 1947), p. 53.
  27. A. C. Crombie, Augustine to Galileo: The History of Science 400- 1650 (London, The Falcon Press, 1952), p. 72.
  28. Abd-ar-Rahman Ibn Muhammad Ibn Khaldun Al-Hadrami, The Muquaddemah’s Ibn Khaldun (New York, Bollingen Foundation, 1958), Vol. Ill, p. 132.
  29. Johannes Baamann, Ibn al-Haitham’s Abhandlung uber das Licht (Leipzig, Halle Als, 1882), p. 37.
  30. George Sarton, Introduction to the History of Science (Baltimore, The Williams and Wilkins Company, 1931), Vol. II, Part II, p. 761.
  31. Sir Thomas Heath, A History of Greek Mathematics (London, Oxford University Press, 1921), Vol. II, pp. 293-5.
  32. W. W. Rouse Ball, A Short Account of the History of Mathematics (New York, Dover Publications, 1960), pp. 161-2.
  33. Mustafa Nazif Bik, Al-Hasan ibn al-Haitham (Buhuthuh wa-Kushufuh) (Cairo, Mutba’ah al-‘timad bi-Masr, 1942), Vol. I, p. 9.
  34. Sarton, Introduction to the History of Science, op. cit., Vol. II, Part II, p. 761.
  35. Rene Taton, History of Science (New York, Basic Books, 1963), Vol. I, p. 482.
  36. Heath, op. cit.
  37. H. L. Kelly, ‘History of Astronomy,’ in Martin Davidson (ed.), Astronomy for Every Man (London, J. M. Dent and Sons 1953) pp. 412-3.
  38. Rom Landau, Islam and the Arabs (New York, The Macmillan Company, 1959), p. 185.
  39. Ball, op. cit., p. 161.
  40. Seyyed Hossein Nasr, Science and Civilization in Islam (Cambridge, Mass., Harvard University Press, 1968), p. 50.
  41. Howard Eves, An Introduction to the History of Mathematics (New York, Holt, Rinehart and Winston, 1969), p. 194.
  42. Sir Thomas L. Heath, The Thirteen Books of Euclid’s Elements (New York, Dover Publications, 1956), Vol. I, pp. 88-9.
  43. Kamal-Addin Abi al-Hasan al-Farisi, Kitab Tangih al-Manazir (Hydera bad, India, Bi Mutba ‘att Majlis da’ irat al-Ma’arif al-‘Uthmaniyah, 1928), Vol. II, pp. 310-20.
  44. Hakim Mohammed Said, ‘Ibn al-Haitham Was a Bridge Between Ancient and Modern Sciences,’ Ibn al-Haitham (Karachi, Pakistan, The Hamdard Academy Press, 1969), p. 29.
  45. Carl Fink, A Brief History of Mathematics (Chicago, The Open Court Publishing Company, 1900), p. 320.
  46. Arnold and Guillaume, op. cit., p. 387.
  47. Francis J. Carmody, The Astronomical Works of B. Kurra (Berkeley, California, University of California Press, 1960), p. 15.
  48. Robert W. Marks, The Growth of Mathematics from Counting to Calculus (New York, Bantam Books, 1964), p. 120.
  49. Indian Office Library, London, England, Arabic MSS, 744, fol. lb-2a.
  50. Florian Cajori, A History of Elementary Mathematics (New York, The Macmillan Company, 1917), pp. 126-7.
  51. Sydney N. Fisher, The Middle East (New York, Alfred A. Knopf, 1969), pp. 116-7.
  52. David Eugene Smith, History of Mathematics (Boston, Ginn and Company, 1925), Vol. II, pp. 455-6.
  53. Charles Singer, A Short History of Scientific Ideas to 1900 (Glasgow, Clarendon Press, 1960), pp. 151-2.
  54. Aydin Sayili, Thabit Ibn Kurra’s Generalization of the Pythagorean Theorem,’ Isis, LI (March 1960), Part I, No. 163, 35-6.
  55. George N. Atiyeh, Al-Kindi: The Philosopher of the Arabs (Karachi, Al-Karimi Press, 1966), pp. 166-8.
  56. Sorbonne University, Paris, France, Arabic MSS, 2544 i bl. Gal. SI. 374.
  57. Florian Cajori, A History of Mathematics (New York, The Macmillan Company, 1919), p. 104.
  58. Indian Office Library, London, England, Arabic MSS, 750, fol. 41b- 42a.
  59. Solomon Gandz, “The Sources of Al-Khwarizmi’s Algebra,’ George Sarton (ed.), Osiris, I (1936), 264.
  60. Solomon Gandz, The Geometry of Muhammed ibn Musa al-Khwarizmi (Berlin, Verlag von Julius Springer, 1932), p. 64.
  61. George Sarton, A History of Science (Cambridge, Harvard University Press, 1959), Vol. II, p. 48.
  62. De Lacy Evans O’Leary, How Greek Science Passed to the Arabs (London, Routledge and Kegan Paul, 1951), p. 158.
  63. Taton, op. cit., p. 408.