After a brief commentary about Haskell Brooks Curry, (born in Massachusetts on September 12, 1900), Dr. Barsky's presentation began to generate an overall theme of Archimedean logic. The topics covered consisted of Haskell Brooks Curry, and Archimedes and proposition I of the scroll called the Measurement of a Circle.

Within the topic of proposition I, the fact that he used similar triangle properties the way he did interested me the most. According to the presentation by Dr. Barsky, Archimedes began this proof by contradiction. He set the circle greater than the triangle k for the first contradiction, and the circle less than the triangle k for the second contradiction. When he determined both original settings to be false, he knew that he would prove the area of the circle to be equal to the area of the triangle k, an excellent example of the logic used in Archimedes time. When he came to the triangular arguments, he finished his proof. I believe that Archimedes learned a great deal about triangles from his predecessors, and the fact that he had built up a large knowledge base allowed him to expand on it. If he had no one around to teach him anything about mathematics, he may not have had so much insight. However he took mathematics to new heights.

Today our discussion picked up where it left off with Archimedes. This class period was devoted to his proposition 1 that relates to the area of the circumference of a circle. Although Archimedes established the fact that the area of a circle is equal to the area of a triangle, his argument does not tell how the triangle is constructed (with compass and straight-edge). Therefore this does not mean that there is a square of a circle.

In the proof of proposition 1, let ABCD be the given circle, and K the triangle described. If the circle is not less than or greater than K, then it must equal it. The three possible assumptions were given a try. It was concluded that since the area of the circle was neither greater than nor less than K, they are equal.

On the Measurement of the Circle (proposition 3 of 3) was one of the most popular of the Archimedian works during this period. Archimedes inscribes and circumscribes a circle by regular polygons of 96 sides. After this, he calculated their perimeters and then assumed that the circumference of the circle was between them (3 10/71 < pi < 3 10/70). This was much more accurate then the Egyptians and the Babylonians.

As is his custom, Dr. Barsky opened the lecture with a comment on this day in math history. Today was the birthday of Haskell Brooks Curry, who was born on 12 September 1900. Curry studied in Germany, taught at Princeton, Harvard, and Penn State, and did a great deal of work with mathematical logic.

The discussion then turned to Archimedes, and specifically focused on his proposition that for every circle, there exists a right triangle whose height is the radius of the circle, and whose base is the circumference of the circle. This triangle has area equal to that of the circle.

There were three problems that kept popping up with the ancient civilizations: doubling a cube, trisecting an angle, and squaring a circle. Presumably, Archimedes' proposition was intended to try to resolve the latter. It turned out that it didn't because the base, 2*pi*radius is irrational. But it did accomplish something else.

Today we know this proposition is true simply from the formulas for area and circumference of a circle and area of a triangle. But back then there was some question as to whether or not the constant, pi, in the formula for the circumference was the same as the constant, again pi, in the formula for area. Archimedes' proof showed that they are the same.

Dr. Barsky illustrated Archimedes' proof. It was based on the tautology that the area of the circle was either greater than, less than, or equal to the area of the triangle. Archimedes assumed that the area of the circle was greater, then, by inscribing polygons and using exhaustion, came to a contradiction. He then assumed that the area of the triangle was the greater of the two areas and found a contradiction using a similar strategy. This left equality as the only possibility.

While reading what Katz has to say about Archimedes, I found it interesting how many widely varied pursuits Katz connects with Archimedes. The problem of equating the circle and a triangle was only one of many. Archimedes also did extensive work with levers and pulleys, with finding the area of a parabolic sector and comparing it with an inscribed triangle, and summing infinite series. I would imagine that by then, people must have considered the idea that use of a lever will make a task easier common knowledge. But Archimedes actually put numbers and measurements into it. He even assumed that the lever was rigid and weightless, just as we do today when teaching this principle. He also made an estimation of pi that was as close as most estimates commonly in use today. I find the genius and ingenuity of this man amazing.

This lecture was entirely about Archimedes' proof concerning the measurement of a circle. The proof was to show that the area of a circle is equal to the area of a right triangle, with the circumference as one leg and the radius as another.

I thought this lecture was interesting because of the way we did the proof in two steps. First we proved the area of the circle was not greater than the area of the triangle, then we showed that the area of the circle was not less than the area of the triangle. This was interesting because I had never thought of the use of formal logic steps to do a geometric proof.

Today's person in history was Curry, who was born today in 1900. He only died 14 years ago. He taught in many places in the US, including Harvard and Penn State. He was Harvard educated and got his doctorate in Germany in 1929.

Our main topic of the day was Archimedes. We discussed his proof of the area of the circle, or squaring a circle. He proved that if you know the radius and the circumference of a circle, then you could figure the area of the circle by using the radius and circumference as legs of a right triangle and finding its area. The proof was a proof by contradiction. First he proved that if you assumed that the area of the circle was greater than the area of the triangle, then you would get a contradiction. Next he proved that if you assumed that the area of the triangle was greater than the area of the circle, then you would also get a contradiction. Thus the only possibility left was that the areas were equal. Archimedes used inscribed and circumscribed polygons to do his proof. It would seem Archimedes had some sort of concept of infinity. In his proof he said you were to draw the polygons as many times as necessary. So, he did seem to realize you could continue to draw them without coming to a point where you had filled the whole space.

Archimedes is also given credit for coming up with a numerical representation for pi, which is used even to this day. So, it would seem that he was able to make the connection between the circumference constant and the area constant being one and the same.

This lecture continued our discussion of Archimedes and his work concerning the measurement of circles. Proposition #1 of Archimedes, which relates the area of a circle to the circumference of a circle, was presented in detail. Proposition #1 states that the area of a circle is equal the area of a right triangle in which the legs of the triangle are the radius and circumference of a circle. Since the area of a triangle is (1/2)bh, this can be written as (1/2)(circumference)(radius), or (pi)r^2 as we know it. Thus Archimedes was aware the relationship between the area and circumference and the circular constant pi. It is known that Archimedes was able to approximate pi fairly closely, and it seems likely that he might have used this relationship between a circle and its corresponding triangle to find a way to approximate pi. The proof of proposition #1 was a fairly complicated geometric proof showing by contradiction that the area of the circle could be greater than that of the triangle, and conversely, that area of the triangle could not be greater than that of the circle. I found the proof of this proposition very fascinating, and find it very interesting how Archimedes' method anticipates the idea of limit.