Throughout my study, I have touched on all of the preliminary questions with which I began my study; however, it is convenient to discuss these questions and conclusions in a separate section. The first observation that needs to be made is the surprising amount of Islamic mathematics which is relevant to contemporary studies, but is not part of any mathematical course. This is vastly apparent even with my three questions that I set out to answer; many of my presuppositions and assumptions were wrong, which made the questions themselves invalid. It is astounding how much historical mathematics is omitted from mathematical courses, which could greatly benefit from the inclusion of even a small amount of this material. In many cases, my understanding of modern mathematics, especially algebra, was enhanced by my study of Islamic mathematics; the historical overview gave insights to modern notation and modern methods of problem-solving.
Algebra vs. Geometry
I began my study with the question: "Why was the geometry of Euclid transmitted verbatim while algebra was created and innovated by Muslim mathematicians? In other words, why was geometry not developed while algebra was both created and developed?" As my research shows, the question itself is not correct. Geometry was developed and added to by Muslim mathematicians, though I began my study without knowing this. Still, geometry only developed so far, and usually developed in conjunction with algebra, so the topic remains one of interest and can still be discussed.
Abū Sahl's explanation of the construction of a regular heptagon shows his innovation as a geometer and his contribution to Islamic mathematics by providing solutions to "impossible" problems within known mathematical theories. Though Archimedes provided a construction of a regular heptagon inside a circle, his construction was not proven and so was more of a proof of the existence of a regular heptagon inside a circle rather than a valid construction. Abū Sahl was able to construct and prove the "impossible" by reducing the problem to the construction of two conic sections. By doing this, Abū Sahl showed that the construction of a regular heptagon belonged to an intermediate class of problems which required at times the use of cubic curves. Abū Sahl was able to reduce a problem into a method which was already accepted, that of conic surfaces, to provide a general solution for the regular heptagon problem. Further, the geometry of Ibrāhīm ibn Sinān shows the Muslim importance of concise proofs and constructions, and he improved upon the methods of the Greeks in drawing the parabola, ellipse, and hyperbola. The extension of geometry into practical spheres was also a development which was not seen in Greek mathematics. The existence of many treatises on the problems facing artisans as they constructed their geometrical artwork shows the intersection between theory and practice in Islamic geometry, and in all Islamic mathematics.
Geometry may not have developed to our modern form, or to 3-dimensional geometry beyond conic sections because of the environment in which Muslim geometers worked. In Islam, it is prohibited to make likenesses of people, and so Muslim artists developed geometrical design over attempting to draw people and objects in 3-dimensions. It is this drawing in 3-d which may have sparked the Western European geometrical development; therefore, the Islamic culture in the House of Wisdom precluded the further development of geometry past the point already discussed.
The Nature of Proofs in Islamic Mathematics
Another question with which I began my study was the following: "Why didn't the Persian mathematicians expand or invent new theorems or proofs, though they preserved the definition-theorem-proof model for geometry? In addition, why did the definition-theorem-proof model not carry over from Greek mathematics (such as geometry) to algebra?" This question was also misinformed from the beginning. Though al-Khwarizmi did not adopt the definition-theorem-proof model, later algebraic treatises do so. Al-Khwarizmi still recognizes the importance of proving all assertions, though he did not structure his theorem-proofs in exactly the same way that Euclid's Elements or other Greek mathematical treatises do. Still, the topic can be discussed even if the question was originally misstated.
Al-Khwārizmī did not, as I had previously assumed, have access to most Greek mathematical treatises. In fact, al-Khwārizmī only had access to a very few parts of Euclid's Elements. His college Thābit ibn Qurra was engaged in the translation of many Greek treatises during and after al-Khwārizmī's publication of Algebra, so al-Khwārizmī must have relied on Hindu and local Syriac-Persian sources for his studies. Al-Khwārizmī did not have access to Greek manuscripts, as seen in his proof of a2 + b2 = c2 which is only valid in the case when a = b. Euclid's Elements provides more rigorous proof which holds for all cases; if al-Khwārizmī had access to this text, he surely would have recognized this and included Euclid's proof, as he included the proofs from Hindu texts of other theorems, such as how to take a square root, when the Hindu proofs surpassed his own. In this case, then, it is an easy answer regarding why the definition-theorem-proof model was not carried over from Euclid to algebra; the founder of algebra was not aware of the existence of the definition-theorem-proof model.
The study of Islamic mathematics, both geometry and algebra, have shown that the definition-theorem-proof model was commonly used after the Greek treatises, and therefore the Greek definition-theorem-proof-model, were available to Muslim mathematics. For example, this can be seen in the already mentioned proofs of Ibn Sinān, who followed this model when discussing the area of a segment of a parabola, or in the discussion of x3 + mx = n by Umar al-Khayyami, which includes a theorem, solution, and proof. It is clear that though it was not considered necessary, most Muslim mathematical treatises included some form of the definition-theorem-proof model that they admired so much in Euclid's Elements and other Greek texts.
The final question I asked before I began this study was the following: "Why were most of the leading mathematicians, in this time period, Muslim? Including, why were there no Jewish mathematicians? Why were there no Orthodox or Arab Christian mathematicians?" This question, too, is flawed; there were Jewish and Christian mathematicians at the House of Wisdom in Baghdad. However, they were employed mainly as translators, as they knew either Hebrew and Arabic or Greek and Arabic. The discussion of this question requires a brief look at the status of Jews and Christians within the Islamic Empire.
The Qur'an accords Jews and Christians the status of "People of the Book." Under Islamic law, then, Jews and Christians are given protected status as "dhimmi". As dhimmi, Jews and Christians are given the freedom to practice their own religion, with the condition that they may not attempt to convert Muslims to their religion, as long as they pay a special tax to the Islamic government. Though this afforded them a protective status, Jews and Christians were thus second-rate citizens; they did not have equal benefits under the law or in court. Further, many Muslims and Islamic governments wished for their second-class status to be reflected in their standard of living and their political and societal prominence.
This may help account for the position of Jewish and Christian mathematicians. Though employed as translators by al-Rashid and al-Ma'mun, these scholars were not seen as equals of Muslim scholars at the House of Wisdom. The formation of Islamic theology was also happening during this period, and many Jews and Christians gained prominence from their positions as religious authorities; they were asked to account for their religion before the Caliph, Islamic theologians, and Islamic legal analysts. However, it seem that it was not recognized that Arab Jews and Arab Christians could be experts in intellectual fields besides religion; at least, it was thought that Muslim scholars could perform just as well in all intellectual fields besides religion, where the caliphs were fair enough to let Jews and Christians speak for themselves. This bias may be recognized in the financial support given by the Caliph to specific Muslim mathematicians to allow them to write original treatises. The caliphs may have been willing to pay Arab Jews and Arab Christians for their translation works, but seemed to favor Muslim mathematicians when requesting original mathematical treatises.
Please see the "Paper" section for further information on these conclusions.