Medieval China

The main topics David covered were a general history of the Chinese dynasties, the beginnings of counting, the area of the circle and Liu Hui's measurement of a sea island. China has gone through many dynasties from -2000 to 1368, each of which is separated by a time of anarchy. The first dynasty was the Shang, during which math was mostly counting. The way they counted was on counting rods, which could be arranged horizontally or vertically. Also there were Oracle bone numerals that were very similar to the counting rods. After the Shang dynasty came the Chou (-481) dynasty. This was a time of barbarians who used the scholars to help them control the people. The Chin dynasty (-221) followed next and was known for burning books. The Han dynasty (-210) was when the formula for the area of the circle was proven using bows to form a parallelogram. The dynasties that followed were Wei (221-581), Tang (591-907), and Song (960-1368).

The main mathematician that was covered was Liu Hui. He lived during the Wei dynasty and wrote a book called "Sea Island Mathematical Manual". David showed us his first problem from this book, which measured the distance to and the height of a sea island. He said that they were not sure how Liu came up with his formula. The two examples David covered were speculations on how Liu might have derived his formula. The first used similar triangles and then set up a ratio that would give you the height. The second way used congruent triangles and subtraction of area to come up with another ratio. In both cases the important point was forming the ratio that could use the given numbers to get the height.

As is his custom, Dr. Barsky started off today's class with a tidbit from "This Day in Math History." One of the people born on this date was Jean Baptiste Joseph Delambre, who made substantial contributions toward devising the measurement system now known as the metric system. Dr. Barsky then turned over the podium to student-lecturer David Sedorook.

Mr. Sedorook proceeded to guide the discussion through the history of Chinese dynasties from around 2000 B.C.E to the fourteenth century of the common era. Mr. Sedorook took time out along the way to talk about some bones found in the early twentieth century, called oracle bones, which contain markings similar to those on the counting rods discussed during lectures on the ancient civilizations. These markings on the oracle bones appear to resemble modern Chinese numbers. Mr. Sedorook also discussed the Han dynasty, which enforced a very imperialistic rule. During the Han dynasty (c.200 C.E.), the Chinese had somewhat of a caste system, where soldiers and government officials were given high status, and agricultural workers were viewed as part of the lower class. During the Han dynasty, China introduced a system of civil service exams. Katz also writes about these exams, saying that while they provided some encouragement for studying math, they provided little incentive for creativity. Instead, they focused on recitation of mathematical texts, and working out a series of standard problems.

Mr. Sedorook spoke briefly about the interest the Chinese had in finding the area of a circle, doing a demonstration of how a circle can be broken up into wedges. By rearranging the wedges, a shape can be created that resembles the shape of a parallelogram. Of course, by using a limiting process as the norm of the wedges goes to zero, we would essentially have a rectangle, but the concept of limits wasn't used by the Chinese at that stage.

Mr. Sedorook then focused his discussion on a problem of how to find the distancce to and height of an island, given sightings along two poles of known height and distance from each other. Mr. Sedorook proceeded to go through a solution method to this problem that was possibly used by the proposer of the problem, Liu Hui, as well as another solution due to Yang Hui.

The Mathematicians of the Day were James Waddell Alexander, William Henry Fox Talbot and Jean Baptiste Joseph Delambre. A brief history of the various dynasties that held power in Western Russia and what we call China today. The evolution of the Chinese numerals was displayed. Also problems were solved out of the Sea Island Mathematical Manual.

When David was going over the evolution of the Chinese numerals I noticed that they lacked a zero symbol or a place holding symbol, although they did acknowledge the 10's places. I found in Katz that a dot was used for zero in a Chinese document dated 718 that was written by an Indian scholar employed by the emperor of the time.

This presentation began with the usual mathematician of the day. Today's mathematician is Jean Baptiste Joseph Delambre. He was the first to measure the length of the arc from Dunkirk to Barcelona. After that we made a quick trip through all of the Chinese dynasties, starting with the Shang. Next was the Zhou dynasty who, while very warlike, they had a strong belief in education, including mathematics. The Qin dynasty built most of the great wall but they burned nearly all of the books to keep the general populace uneducated to keep them under control. Then came the Han dynasty who were strict rulers. They started a class system but they brought back mathematics. Then we discussed problems from some of the Chinese texts.

The most interesting thing I found about this class was how the Chinese were so interested in using mathematics for surveying. It could not be a coincidence that they build the great wall. Also most of the Chinese texts had extensive treatments of surveying. With all of the wonderful archetecture of the land they obviously made some wonderful discoveries in this subject. Also, the Chinese were not thwarted by irrational numbers like the Greeks were. This made for some bold discoveries with finding the areas of circles and semicircles.

David's topic of mathematics in Medieval China began with the following of the ancient dynasties of China up to the Medieval age. As he moved from one dynasty to another, he gave a brief explanation of what was happening in the field of mathematics during each dynasty. Finally, he did a few Chinese math problems involving area of a circle and distance to and height of an island.

I thought the most interesting thing was the area of the circle. After dividing the circle into 'pie slices' or "bows", each piece was set next to another with alternating pieces being inverted. Once all the bows were in place, they made a rough form of a parallelogram. Using the area of a parallelogram, the circle's area could be calculated to a certain degree of accuracy.

The presentation began with a brief history of early China and a description of the earliest Chinese counting systems, which began with the system of counting rods. They consisted of vertical or horizontal lines arranged to represent the numbers 1-10. The next system consisted of a combination of the lines for 1-4 and a set of symbols for the remaining numbers. A long history of the many dynastic changes and wars was presented, but it was really too much information to absorb. Some of the highlights include the burning of books in the Chin dynasty, who began the building of the great wall, sort of a cultural revolution part 1. Also important was the introduction of the civil service examinations under the Han dynasty. The Han encouraged scholarship and the merit based civil service allowed talented individuals to pursue subjects such as mathematics.

A problem from the Han dynasty was presented in which the formula for the area of a circle was approximated by separating a circle into many wedges, called bows, and arranging them in an alternating pattern in order to form a rough parallelogram. The area was then approximated to be the circumference times the diameter, which corresponds to the base of the parallelogram times the height.

A problem from the third century was presented next. The problem comes from a book titled The Sea Island Mathematical Manual, written by Liu Hui in the third century. The problem concerns finding the distance and height of an island. The presentation was a little hard to follow, but it had to do with setting up a system of parallel lines and similar triangles, setting up a ratio of distances and solving for the height and distance. Another method was shown in which involved constructing a rectangle from the height and the distance and using the areas of congruent triangles to solve the same problem.