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Chapter 2, Part 2
In 622 A.D. the prophet Mohammed, fearing for his life, fled Mecca for Medina. In less than ten years, he was to return triumphantly as the military head of an Arab state. This is about the time that Brahmagupta of India was compiling the many and varied mathematical achievements of the Hindu people. He shows us evidence of geometric and number theoretic thought, and even work with quadratic equations having negative terms and roots . The East then would be another source for the Islamic learners, once they had had their fill of grabbing territory, and began collecting wisdom instead. Mohammed fell even faster than he rose, dying suddenly as he was preparing to launch a conquest of Byzantium in 632.
After the death of Mohammed, a series of four Caliphs (translated, deputies) carried out his instructions to convert all people to Islam by force. They waged war with their armies numbering in the thousands. They moved westward against Damascus, Egypt, Carthage, and ultimately, across to Gibraltar and Spain. To the north, they made several challenges to the Byzantine Empire with intermittent success. Their conquests extended eastward through Persia and into India. This is not the only example in history of bloodshed fueled by religious fervor, but it is one of the swiftest and most expansive.
If the Arabs had presented a unified front, they may have been able to conquer most of Europe which was weak at the time. They instead fought amongst themselves all during their expansion. There were disagreements over the succession of caliphs, which resulted in a permanent split between the Sunni of Syria, and the Shi'a of Persia. After the death of the fourth Caliph Ali, there were no more leaders from Mohammed's family, and so the regional kings began vying for power. Mu'awiya of Syria, the next ruler of the Damascus Caliphate, began the Umayyad Dynasty (661-750) and continued the westward expansion. Not all the caliphs were wont to present the warrior role model for their armies. Those leaders spent most of their time in their harems and left the fighting to underlings. As a result, in 750 the Umayyads were challenged and defeated by the Abbasids, whose dynasty marks the Golden Age of Islam.
Meanwhile, the armies had marched through Spain and into southwestern France. They were met at Poitiers by combined armies under the direction of Charles (the Hammer) Martel. The Arabs retreated after the first day's battle, and they never challenged for more territory after that. The empire was as big as it could possibly afford to be. Spain would wait three hundred years, however, to be liberated. For more details on this conquest, I recommend Boyer  and especially Wilson , who makes a vivid account.
One place that made the biggest impression on the Arabs, as they were marching westward, was the city planned by, erected for, and named after Alexander the Great. Beyond the visual spectacle of gleaming white marble edifices, there was the grand university which was a repository for classical culture and knowledge. Word got back to the harems, and the envious caliphs, especially al-Mansur in 762, attempted to emulate Alexandria at Baghdad. A brick wall and 160 towers were erected for protection, and scholars came, bringing the learning extant from far-flung places, but especially from India and Greece, by way of Persia, Alexandria, and Constantinople .
Baghdad's learning center was to be known as The House of Wisdom, and it lived up to its name. Many of its mathematical advances centered around astronomy. In Greek culture, the prediction of an eclipse would garner admiration for the intellect of the harbinger. We are now at a time in history when queasy monarchs cultivated learning in order to maintain the respect and fear of their subjects and enemies. This would be the case in Europe as well, and it would stay that way for hundreds of years. Making horoscopes for royalty, for example, is how Kepler earned his subsistence in the 17th century .
During this flowering in Baghdad, there was much improvement in trigonometry and spherical geometry, and of course, they continued to develop their strength, which was algebra. More important possibly, was the synthesis which united the mathematical branches and the many cultural standpoints. By order of the Caliphs there was a rush to translate the Greeks, especially Euclid and Archimedes, and there were also translations made of the works from Hindu lands. Scholars began to see Euclid, the aha problems from ancient times, number theory from the east, and modern applications as a unified discipline. One translator in particular should be mentioned. Qurra, working in the late ninth century, improved on and extended the Arabic translations of Euclid, Archimedes, and others. He and his colleagues are credited with developments in language which were sophisticated enough to describe the rational processes. This is a period when the use of many word-forms, stemming from a common root word but possessing shades of meaning, was invented. In some cases, the Arab translations are the sole source of works by Archimedes .
A 10-lattice based on the Golden Ratio..........and a lattice combining 4, 7, and 8-way symmetry.
One who is given most credit for the synthesis at this time was al-Khwarizmi, an astronomer who worked at Baghdad and Istanbul. In 820 he published a book with aha problems, and a way to represent them with variable equations, along with methods for manipulating those equations toward solutions. In the title of the book were the words "al-jabr" meaning completion, and referring to augmentation of both sides of an equation, and also "al-muqabala" meaning reduction, and referring to the inverse process of ridding both sides of duplicated terms. We still refer to this branch of mathematics as al-jabr or algebra. Al-Khwarizmi also lent his name for the word we use today to describe a method of solution: "algorism" (later algorithm) .
It certainly cannot be claimed that al-Khwarizmi invented any of the mathematics for solving quadratics. The necessary geometry is found in Euclid, the exemplary problems solved on Babylonian tablets, and as mentioned, much of the theory is there in Hindus writings. Al-Khwarizmi put the manipulations of the terms of the equations in juxtaposition with corresponding geometric constructions. As a result, mathematicians of his day, especially geometers, became more confident in equation handling. Later, when Europe rediscovered Euclid, Latin translations of al-Khwarizmi's works were the most influential. It seems that he arrived at the "teachable moment." 
Two cases illustrate this synthesis. Al-Khwarizmi actually used five separate cases to avoid the negative terms which were anathema to geometers. In Figure 1, we see the construction which is well known as "completing the square." Keep in mind that the purpose of these constructions was to lend support to the numerical manipulation of the coefficients and terms. In this first case, the sum of the first and second degree terms is a constant. The square of half of the linear coefficient 10 is the amount, added to both sides, which completes a square .
In the next case, the square of the unknown equals the linear term augmented by a constant. The Hindus, working with the negative term and applying the rules of integer arithmetic, would have transposed the linear term and completed the square. Al-Khwarizmi brings a construction method into agreement by placing the areas 5x and 14 inside the square. The segment BE (coefficient 5) is then bisected, and its square EHKM augments the 14 by a clever shift of area WNFD. The sum is contained in square SHAW [see Figure 2] .
The Islamic cultivation of knowledge in the ninth and tenth centuries became an important link in the advancement of mathematics. In what amounted to exercises in counter-espionage carried out over 400 years or so, the "borrowing" of works from antiquity, the subsequent Arab translations, and the ultimate "borrowing back" accomplished by Latin scholars in the twelfth century, became a bridge of preservation for Western culture .
The Moslems neither initiated nor completed a theory of equations. They assembled the contributions of scientific theory and practice from the numerous peoples that they conquered in their expansion phase. They did lay the foundation for Omar Khyyam's cubic solution and for Cardano and other Renaissance mathematicians who would solve the general third and fourth degree equations. Many regard Abel (Norwegian, 1802-1829) as the one who completes the theory by showing there is no solution by radicals of equations of fifth or higher degree .
Some Islamic mathematical advances were of immediate benefit to astronomers who were trying to improve on the tables left by Ptolemy, those which predicted celestial events. Arab astronomers made marked improvements in the detail and accuracy of predictions, however, there was a limit to the accuracy they were going to achieve, using the wrong model for our planetary system. The discoveries of Copernicus, Kepler, and Newton would not merely set the orbits in proper motion, but also would give us new analytical tools such as the calculus. Area would be an important concept once more, this time in connection with the conic curves. You will find more details in Chapter 3 of this project.
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