The presentation given today by Dr. Barsky, after briefly commenting on the birth of Charles Sanders Pierce (9/10) and the death of Gabrielle Du Chatelet (9/10), held to the main concept that mathematics should be looked at from an internalistic point of view. The topics covered consisted of Charles Sanders Pierce, Gabrielle Du Chatelet, Euclid, Ken Pledger, Proposition II.11, and the introduction of Archimedes.
Among these topics, Proposition II.11 interested me the most. I thought Euclid used his powers of rationality well on this one because he came up with the Pythagorean Theorem again, even though he may not have realized it at the time. According to the presentation, he started the solving sequence by cutting a line segment in mean and extreme ratio, which led to a quadratic equation x^2 + ax - a^2. When Dr. Barsky solved this equation, he came up with a positive solution of x equals the sqrt((a/2)^2 + a^2) - a/2. Now sqrt((a/2)^2 + a^2) gives us the hypotenuse of a triangle, and x equals the segment AF. According to Dr. Barsky, we can think of Euclid's geometry as dealing with objects that can be created using compasses and straight-edges. In my interpretation, I think that Euclid had some type of geometric shapes in front of him so he could see that the area of one object equals the area of another object, as well as being able to see one segment being equal to another. Although he may have known some methods, I believe he still had to play around with the figures and the formulas.
Today's lecture started with Dr. Barsky telling us that today is the birthday of Charles Pierce. Pierce was mostly a researcher, spending only about five years in academia. He did substantial work on the Coast and Geodetic Survey, and spent considerable time on the graph coloring problem.
The next topic of discussion was Euclid's proof of the existence of golden ratios. Dr. Barsky mentioned a letter written to the math history list which commented that Euclid's proof uses letters to represent points. This is opposed to the ancient Egyptians, who would use numbers and assume that the reader could generalize by plugging in their own numbers and follow the same procedure. Euclid was among the first to use variables, and even Euclid's variables represented points on a plane, not numbers. Dr. Barsky then went through Euclid's proof, and commented that all of Euclid's geometry appears to use some geometric algebra.
Finally, we talked about Archimedes. Archimedes was a mathematician, physicist, and engineer. He did a lot of work with pulleys, catapults, and levers, basically making a mathematical model to represent a lever. He also theorized on the reflective property of parabolas. This I found amazing. It has only been in recent years that this property has been put to serious use, with car headlights and television satellites, among other things. But we've had knowledge of this property for over two thousand years!
I also found the discussion on the use of variables interesting. In my eyes, this practice of abstaining from using variables is amazing. Using variables boils down to nothing more than representing one thing with another. Today, young children use objects to represent people or other objects in backyard games. They diagram football plays using a rubber band or paper clip to represent a particular child. This is the same principle as using variables. But, despite all the incredible ideas these great mathematicians came up with, the idea of using symbols to represent unknown numbers, or generalize unknown quantities never came up.
The discussion on golden ratios has brought a question to my mind, as well. Why, I wonder, were these people so fascinated with golden ratios? Were they believed to have some connection with the gods? Katz almost skips golden ratios altogether, and the question of why has never really been addressed in class.
Today we talked about Euclid. He is very difficult for me to understand, so fortunately, we received the "royal road" version of proposition II.11. In Euclid's book II, we get geometrical proofs of the distributive and commutative laws (of multiplication) and a geometrical (rectangles and squares) solution to a(a-x)=x^2. After tracing steps from my notes and following this proposition in A History of Mathematics (Boyer), I realized that I am actually able to make head and tails of these.
The figure used in the proposition I just mentioned is used a lot to discuss the iterative property and the golden section.
We began discussing Archimedes. You mentioned the fact that he wrote in a different manner than Euclid. He wrote small works on specific problems that he had solved and shared these with friends (other mathematicians), he also discovered how to find the area in a parabola. Today, we would use calculus for this.
Today we were told about Charles Pierce and his research on the four color map problem. Also we were introduced to Gabrielle du Chatelet; she wrote a translation of Newton's "Principia" into French. The main topic of the day was Euclid's Proposition 11 from Book II of the Elements.
It was interesting that Euclid first used variables to identify shapes, such as AB=line, ABC=triangle, etc. Euclid's style of writing became the mold for writing by mathematicians. Euclid used geometry to prove algebraic concepts.
Archimedes was also discussed. He was known for his engineering abilities. He developed levers and pulleys for use in his country's defense against Roman ships.
This lecture began as a continuation of our discussion of Euclid's Elements. Proposition 11 from Book II was presented in detail as an example of Euclid's method of geometric proof. Euclid's use of variables to represent arbitrary points, as opposed to representing the magnitude of a line segment, for example, was mentioned as perhaps being the introduction of the use of variables in Greek mathematics. Replacing Euclid's variables with algebraic variables representing lengths, it was shown that the length of the segment X, on which the square was formed, is the solution to the equation X^2 + aX - a^2 = 0, where a= the length of the whole segment. It was then shown how the Pythagorean theorem can be related to the structure of Euclid's proof, raising the question of a Babylonian, or algebraic influence in his geometric proofs.
The lecture concluded with a brief introduction to Archimedes, who lived in Syracuse in the -200's. Archimedes was a brilliant engineer as well as mathematician. Archimedes' mathematics were less formal than Euclid's, but his methods were similar in that they relied on geometric logic to deduce his propositions. He worked on the relationship of diameter to circumference and area, and approximated the circular constant pi.
This lecture continued our discussion of Greek mathematics, apparently dealing more with Euclid's time and the middle period of the long and distinguished span of ancient Greek intellectual leadership.
But at the beginning of class, as usual, we learned about a few important mathematicians connected with the current date--namely, Charles Sanders Peirce and Gabrielle du Chatelet. The former, born the son of Harvard math professor Benjamin Peirce on September 10, 1839, worked for the Coast and Geodetic Survey, taught at Johns Hopkins University, and worked on the Four-Color Problem. The latter was married to a marquis, worked with Voltaire on a French translation of Newton's Principia, and died on September 10, 1749. We then heard a brief quote lamenting the fact that few mathematicians take the time to stop and smell the Elements, so to speak, and making the surprising proposition that mathematical rigor is a subjective and relative concept. But most of the lecture centered on Euclid's treatment of his Proposition 11, Book II.
This showed how one could "cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment." Briefly, this meant that a line segment AB could be divided at a point H into segments AH and HB in such a way that the area of a rectangle with dimensions HB by AB would be the same as a square with dimensions AH by AH. This was done by first constructing a larger square with the segment AB on one side, then bisecting the adjacent side AC at E and connecting the far corner B to E to form the hypotenuse of a right triangle with sides AE and AB. Then by the Pythagorean Theorem, the square of AB plus the square of AE (which = 1/2 * AB) equals the square of EB, which length can be shown to equal the sum of AH and AE. But to find the exact point H along AB means solving this quadratic equation for AH, which the Babylonians had learned to do centuries earlier by "completing the square". Euclid does this geometrically by composing the smaller square with sides of length AH measured by subtracting the length AE from a line segment of length EB extending from E through A to F. In making this "Golden Section" he relies on Proposition 6 and basically solves the same quadratic equation generated by Proposition 30 of Book VI. In this, he finds the "extreme and mean ratio", or the "Golden Ratio", which is AB:AH = AH:HB = [sqrt(5) + 1] : 2.
I hope what we got from this lecture is an approximate feel for the nature of Euclid's Elements and maybe an appreciation for the sophistication of the ancient world's grasp of two-dimensional mathematics. Personally, I got as much insight from the point made early on in the period comparing the relative rigor, subjective certainty, and degree of reliance on others' results of mathematicians in Euclid's time versus those of our own.