The presentation given today by Dr. Barsky generated an overall theme that the works of the mathematicians of Islam during the period of (965-1039) seemed to show traces of calculus, despite the fact that calculus came about later in time. In this lecture, the topics consisted of al-Khwarizmi (750-850), Thabit ibn Qurra (830-890), Abu-Sahl al-Kuhi (early 900s), ibn al-Haytham (965-1039), Mohammed's Flight from Mecca (622), the Battle of Tours (732), the period of Caliphates, the Fall of Baghdad to Seljuk Turks (1055), the beginning of the first Crusade (1096), the arrival of the Mongols under Ghengis Khan (early 1200s), al-Khwarizmi's truncated pyramid problem, Mishnat ha-Middot, and ibn al-Haytham's volume of a paraboloid calculations.
Among these selected topics, al-Khwarizmi's truncated pyramid problem interested me the most. Previously, Dr. Barsky had covered this during the part of the Ancient Mathematics chapter on the ancient Egyptians. In that chapter and lecture, the possibilities for calculation consisted of either extending the truncated pyramid to a full pyramid and subtracting the volume of the top portion from the volume of the full pyramid to get the volume of the truncated pyramid, or breaking the truncated pyramid into several smaller regular pyramids and adding up their volumes to calculate the volume of the truncated pyramid. Later, during the period of Islamic mathematics, al-Khwarizmi used the first of these two methods, and, unlike the Rhind Papyrus, clearly explained his method. In the Islamic truncated pyramid example, al-Khwarizmi set up the equation (the volume of the full pyramid equals the volume of the truncated portion plus the volume of the top portion). To solve this equation, he would have to subtract the volume of the top portion from the volume of the full pyramid to find the volume of the truncated portion. He could calculate the volume of the full pyramid and the volume of the top portion by using the (1/3Bh) formula. This would allow him to find the volume of the truncated pyramid algebraically. I think that this may be an excellent example of how the Islamic people may have come in contact with the knowledge of the Greeks, and then expanded on it by taking their link between algebra and geometry and refining it. The Islamic mathematicians actually may have used a formula which they backed up geometrically, and they then made a note of the formula which they could use for later use.
We did not have a mathematician of the day, instead we talked about our papers that will be due October 8. We then covered more of the history of Islamic mathematics. We focused on al-Khwarizmi (750-850), Thabit ibn Qurra (830-890), Abu-Sahl al-Kuhi (early 900s), and ibn al-Haytham (965-1039). We concentrated on problems from al-Khwarizmi and ibn al-Haytham, which mainly dealt with finding volumes.
I really enjoyed the lecture about the volume of a parabola from ibn al-Haytham. I have previously seen the symbol for (the sum of), but I usually stopped at that point because I did not understand, or I felt like it was too complicated. I understand the logic behind finding the circumscribed volume, and inscribed volume of the parabola. I can see that the difference between the two is the volume of the bottom disk of the circumscribed volume. I see that the (sum symbol) is included in equations that are interested in finding the sum of the differences between two estimates which will give you the solution to a problem. I also see that as you take the sum of the differences you are reaching the limit which is related to the volume of the parabola.
I feel that now, when I am confronted with learning this new language (that's about what it amounts to me) of sums, I will be able rise to the occasion without some of the fears I have had in the past.
Dr. Barsky started off today's lecture with an announcement that the math history list is back in operation. He then went directly into the lecture with no further comments on this day in math history.
As a means of setting the stage for the rest of the lecture, Dr. Barsky wrote down the life spans of four Islamic mathematicians and gave a brief overview of what was happening in the world at the time, noting, among other things, Mohammed's flight from Mecca, the battle of Tours, and the beginning of the first crusade. He then discussed al-Khwarizmi.
In particular, Dr. Barsky read the introduction to al-Khwarizmi's most important writing. Briefly, in this introduction, al-Khwarizmi states that his writing is a summary of the methods of al-Muqabala and al-Jabr, the latter of which is the basis for the word algebra. Al-Khwarizmi states in this introduction that his work will discuss "what is easiest and most useful in arithmetic" (Katz, pg. 229). Dr. Barsky quickly went through al-Khwarizmi's method of finding the volume of a truncated pyramid, which entailed finding the volume of a full pyramid and subtracting off the top part.
Dr. Barsky then talked about the Islamics mathematicians' work to find the volume of a paraboloid. He showed how to find the actual volume through integration, then discussed the method of al-Haytham, which involved the use of inscribed and circumscribed rectangles. Dr. Barsky also showed the similarity between the sums, namely that the first inscribed rectangle is equal to the second circumscribed rectangle.
One thing that Katz talked about that I would have liked to hear more about in lecture was the way the Islamic mathematicians dealt with incommensurability. In a lot of areas, the Islamic mathematicians followed the work of the ancient Greeks. But Katz states that "Islamic algebraists early on began to use irrational quantities in their work with equations..." (pg. 251). So while in a lot of ways they seemed to go along with the work of the ancients, the Islamic mathematicians apparently realized that numbers are continuous as well as magnitudes.
Another item in the reading that caught my eye was that the Islamic mathematicians used trigonometry to find the angle to which someone should turn to face Mecca when praying. It seems that in almost every section we run across an example of how people used mathematics in some seemingly unrelated field. Be it art, music, or religion, mathematics seems to pervade every facet of life. This is probably not true of any other discipline, and to me, this is what makes the study of mathematics so interesting.
Al-Khwarizmi was discussed in Mathematics of Islam, part 1, and again in this portion. He made great contributions to this mathematical world we have today. At the beginning of the semester, we looked at the problem of finding the volume of a truncated pyramid (Moscow Papyrus, problem 14). The wording of this problem was difficult to understand. You had to make sure you had the right interpretation. Al-Khwarizmi's approach to this problem actually leads you through his thought process. By the end of the problem, you can clearly see that he uses subtraction. Unlike before, we tried a method of addition before we saw the one of subtraction.
Calculating areas by exhaustion was talked about next. The fact that Islamic mathematicians dealt with such a topic should clearly lead us to believe that they understood the works of the Greeks.
Unaware of the fact that revolving a parabola around its axis had been calculated already, Thabit ibn Qurra found a long and difficult formula to rotate solids and revolutions. It was ibn al-Haytham who realized that methods could be used to revolve a segment of a parabola about a line perpendicular to its axis. He later proved that the volume of a solid formed by rotating x = ky^2 around x = kb^2 is 8/15 of the volume of the cylinder; the latter volume is pi k^2 b^5. We first used calculus to verify this, and then followed al-Haytham. He sliced the cylinder into n disks. He used inner and outer approximations. The relation between the disks was that the V(Inscribed sum) < V(paraboloid) < V(Circumscribed sum). Since al-Haytham already knew formulas for the sums of integral square and integral fourth powers, he used them.
I found this to be quite interesting because in second semester calculus, I had a difficult time finding the volume of, say, a parabola rotated about a line perpendicular to its axis.
This lecture continued our discussion of Islamic mathematics from the previous class period. The focus this time was on the methods used to find the volume of certain geometric shapes. The first example presented was that of a truncated pyramid. We first saw this problem in our discussion of the Egyptians. Unlike the Egyptians, the mathematician al-Khwarizmi, who gave algorithmic approaches to solving practical problems in his text on "al-jabr," presents a systematic approach for solving this problem. If the dimensions of the base, top and height of a truncated pyramid are known, the ratio of one side of the top to one side of the base can be used to tell you how much needs to be added to the height to complete the pyramid. Subtracting the volume of the "top" from the volume of the completed pyramid yields the volume of the original truncated pyramid. While this was obviously not an original solution to this problem, al-Khwarizmi's method shows a move towards a more systematic approach toward solving general problems in order to create formulas applicable to any problems of a certain nature.
The next problem presented was finding the volume of a paraboloid, or a parabolic segment revolved around an axis. The Moslem mathematicians were aware of the method of exhaustion used by Archimedes to prove various geometric properties and using their algebraic techniques were able to formalize this approach. Using this technique, ibn al-Haytham was able to show that the volume of the paraboloid formed by rotating the parabola x = ky^2 around the line x = kb^2 is equal to 8/15 the volume of the cylinder containing the solid with r = kb^2 and h = b. He did this by slicing the cylinder into n disks of thickness b/n. The volume of the solid would then lie between the volume of the sum of the disks inscribed by the solid and the volume of the sum of the disks containing the solid. The limit of these volumes, as n increased towards infinity, would then be the volume of the solid. It is very interesting how these techniques anticipate the development of calculus by hundreds of years, and how much of later European mathematics must have been influenced by their work.
Dr. Barsky began the presentation on The Mathematics of Islam by listing the four mathematicians he would cover, and then giving the class a time line of events in the Islamic world during this period. The first mathematician investigated was al-Khwarizmi (750-850). Dr. Barsky covered al-Khwarizmi's problem of finding the volume of a truncated pyramid. Dr. Barsky noted that al-Khwarizmi's book purported to be a collection of practical ideas for practical uses, but rarely were any of these ideas "practical." In fact, the book notes that in one of his problems, al-Khwarizmi has the solution of a problem shown as x=1/2, where x is the number of humans. Surely this can't be too practical. Next, Dr. Barsky discussed the mathematics of ibn al-Haytham (965-1039). Investigated was al-Haytham's idea for finding the volume of a parabola rotated about a line perpendicular to its axis. The method used is similar, although not exactly the same, as our method of using a definite integral for finding the solution. Al-Haytham simply calculated the value of circumscribed discs, and then inscribed discs, and used an inequality between the two volumes to bound the actual volume. As the number of disks grows, the estimated volume approaches the actual volume.
This may be a bit corny, but to me one of the most interesting aspects of the Mathematics of Islam is the similarity between the shape of middle eastern temples and the shape of a parabolic segment rotated on an axis, as shown on p. 253. It seems that I have seen this shape many times in both mid-eastern architecture, and Russian architecture. Also, I was fascinated by this treatment for finding the volume of the parabola rotated on an axis. The process is so close to integration, and uses such similar methods as modern integration. It seems that this work, especially when compared to all other works in this time period, was especially forward looking. Maybe if al-Haytham hadn't have had to fake insanity for most of his life he could have brought calculus to us 600 years earlier, and from the integration side rather than the differentiation side.