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THE MATHEMATICS OF Ancient Arabs. Firas Hindeleh Grand Valley State University Math In Action February 9, 2013. TIMELINE. 622 Hegira 630 Conquest of Mecca 637 Conquest of Jerusalem 642 Persia conquered 643 Alexandria occupied 647 Tripoli conquered 642-652 First Arab-Khazar war.
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THE MATHEMATICS OF Ancient Arabs Firas Hindeleh Grand Valley State University Math In Action February 9, 2013
TIMELINE • 622 Hegira • 630 Conquest of Mecca • 637 Conquest of Jerusalem • 642 Persia conquered • 643 Alexandria occupied • 647 Tripoli conquered • 642-652 First Arab-Khazar war • 664 Conquest of Kabul • 670 Fortress of Kairouan • 711 Crossing of Straits of Gibralter • 712 Conquest of Sind • 722-737 Second Arab-Khazar war • 732 Battle of Tours
TIMELINE • 661 Umayyad Dynasty founded in Damascus • 750 Abbasids overthrow Umayyads • 754-775 Abdallah ibn-Muhammad al-Mansur, caliph • 762 Baghdad founded • 786-809 Harun al-Rashid, caliph • 813-833 Abdullah al-Mamun, caliph
Muhammad al-Biruni(973-1055) You know well… for which reason I began searching for a number of demonstrations proving a statement due to the ancient Greeks … and which passion I felt for the subject … so that you reproached me my preoccupation with these chapters of geometry, not knowing the true essence of these subjects, which consists precisely in going in each matter beyond what is necessary … Whatever way he [the geometer] may go, through exercise will he be lifted from the physical to the divine teachings, which are little accessible because of the difficulty to understand their meaning … and because of the circumstance that not everybody is able to have a conception of them, especially not the one who turns away from the art of demonstration.
Muhammad al-Biruni The number of sciences is great, and it may be still greater if the public mind is directed towards them at such times as they are in the ascendancy and in general favor with all, when people not only honor science itself, but also its representatives. To do this is, in the first instance, the duty of those who rule over them, of kings and princes. For they alone could free the minds of scholars from the daily anxieties for the necessities of life, and stimulate their energies to earn more fame and favor, the yearning for which is the pith and marrow of human nature. The present times, however, are not of this kind. They are the very opposite, and therefore it is quite impossible that a new science or any new kind of research should arise in our days. What we have of sciences is nothing but the scanty remains of bygone better times.
Achievements in Geometry • Applications of theoretical geometry • New constructions and new theorems • Thorough discussion of the parallel postulate • Extension of Archimedes’ techniques to new solids • Further study of incommensurables, leading to new understanding of “number”
Achievements in Trigonometry • Use of all six trigonometric functions • Determination and use of various trigonometric identities • Simplification and organization of spherical trigonometry • Proof of central theorems in plane and spherical trigonometry • Standardization of techniques for solving triangles • Improved accuracy of trigonometry tables • Applications of trigonometry in the heavens and on earth
Al-jabr and Al-muqabala Al-jabr can be translated as ``restoring'' and refers to the operation of ``transposing'' a subtracted quantity on one side of an equation to the other side where it becomes an added quantity. Al-muqabala can be translated as ``comparing'' and refers to the reduction of a positive term by subtracting equal amounts from both sides of the equation. Thus, the conversion of 3x+2=4-2x to 5x+2=4 is an example of al-jabr while the conversion of the latter to 5x=2 is an example of al-muqabala.
Al-Khwarizmi’s Algebra (c. 825) That fondness for science, by which God has distinguished the Imam al-Ma'mun, the Commander of the Faithful, that affability and condescension which he shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties, has encouraged me to compose a short work on calculating by al-jabr and al-muqabala, confining it to what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, law-suits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned.
Real-world problemAl-Karaji (d. 1019) Of two travelers going in the same direction, the first goes 11 miles per day, while the second, leaving five days later, goes successively each day 1 mile, 2 miles, 3 miles, and so on. In how many days will the second traveler overtake the first?
Al-Khwarizmi’s Geometry In any circle, the product of its diameter, multiplied by three and one-seventh, will be equal to the circumference. This is the rule generally followed in practical life, though it is not quite exact. The geometricians have two other methods. One of them is, that you multiply the diameter by itself, then by ten, and hereafter take the root of the product; the root will be the circumference. The other method is used by the astronomers among them. It is this, that you multiply the diameter by sixty-two thousand eight hundred thirty-two and then divide the product by twenty thousand. The quotient is the circumference. Both methods come very nearly to the same effect.
Al-Khwarizmi’s Geometry The area of any circle will be found by multiplying half of the circumference by half of the diameter, since, in every polygon of equal sides and angles, the area is found by multiplying half of the perimeter by half of the diameter of the middle circle that may be drawn through it. If you multiply the diameter of any circle by itself, and subtract from the product one-seventh and half of one-seventh of the same, then the remainder is equal to the area of the circle.
Abu Kamil’s Pentagon Construction • And if it is said to you: A quadrilateral ABGD is given, whose sides are equal, each of whose angles is right, and each of whose sides is equal to 10, in which we shall inscribe an equilateral pentagon in this way.
Muhammad Abu al-Wafa’ al-Buzjani(940-997) Book on the Geometrical Constructions Necessary to the Artisan A number of geometers and artisans have erred in the matter of these squares and their assembling. The geometers [have erred] because they have little practice in constructing, and the artisans [have erred] because they lack knowledge of proofs.
