A Profile in Genius: Archimedes

What is a Genius? 

A "genius" is a difficult thing to define. I believe there are geniuses in many different facets of life: there are musical geniuses, there are literary geniuses, there are philosophical geniuses, and I would even argue that there are geniuses who have simply perfected how to make an awesome sandwich. I think everyone is a sort of "genius," since we all have the capacity to make life-changing discoveries and we are all able to make some sort of impact in some way.
One person who I consider to be a mathematical "genius" is Archimedes. A brilliant mathematician, his influence is still felt today, since we use much of his area and volume conjectures (more on that later) and we even use some of his inventions, such as the Archimedean screw. In this case, he has definitely shown his capacity to change the world, and I am completely fascinated by him. I would like to take this time to share my (probably shallow) knowledge of this fantastic mathematician with you all.
Archimedes was born in Syracuse, and was able to form an intimate relationship with the king of Sicily, Hieron II. This relationship began when Hieron asked Archimedes for help in getting water out of a boat, which led to the development of the Archimedean screw. This device is still used today in irrigation methods. Archimedes was also asked to figure out if Hieron's crown had been made out of less gold than he had specified, and if he had been cheated by the person who made it. Archimedes allegedly was sitting in the bathtub when he thought about how he could place the crown in water to see how much weight it displaced, thus being able to figure out if it was pure gold or not. This apparently led him to running through the streets, naked, screaming, "Eureka!"
True or not, this tale is a good indicator of Archimedes' mind. While he was incredibly intelligent and way ahead of his time, he spent much of his time with his head in the clouds, thinking about the various conjectures and proofs he was working on. This is especially true in the tale of his death: After several attempts at an invasion, many of them thwarted by Archimedes, Marcellus made his way into Syracuse. Marcellus, however, had great respect for Archimedes, and he asked one of his soldiers to bring the mathematician to him. When the soldier showed up to retrieve Archimedes, the scholar apparently told the solider to wait, since he was working on something. Enraged, the soldier drew his sword and killed Archimedes. Then, Archimedes till the very end devoted all of his thought to mathematics, which is perhaps one of the reasons why he was such an amazing thinker.
One of the things that amazed me most about Archimedes was his development and use of the double reductio ad absurdum. Essentially, the argument is this: Suppose we have two quantities, A and B, that we want to prove are equal. Archimedes presented two cases: (1) the case where A>B and (2) the case where A<B. Archimedes would then prove that neither of these cases could happen, and this would lead to A=B being the only logical conclusion. This, then, is how he proved his famous area of a circle theorem.
I have outlined the proof below since, to me, it is incredibly fascinating. Please do note that these are in my words, however. I also drew some inspiration for both my synopsis on Archimedes' life and for my proof based on this book by William Dunham, which is an interesting read that discusses several different mathematical scholars and their respective proofs.
Before I outline the proof, note that the Greeks did not define Pi in a numerical term. So, Pi was deemed to be the ratio of a circle's circumference over its diameter, or Pi=C/D. This, then, also means that C=PiD, which we will use later on in the proof.

Archimedes' Proof of the Area of a Circle 

Proposition. The area of any circle is equal to a right-angled triangle in which one of the sides about the right triangle is equal to the radius, and the other to the circumference of the circle. 

Proof. We are given a circle with center O, radius r, and circumference C; and a right triangle having a base with length C  and a height of the length r (see Figure 1). We denote A as the area of the circle, and T as the area of the triangle. We will prove that A=T. In order to do this, we utilize a strategy called double reductio ad absurdum, in which we examine the other two cases, or the case where A>T and the case where A<T, and we will eliminate the, showing that the only possibility is that A=T. 

Figure 1

(1) First, we examine the case where A>T. So, the circle's area is larger than the triangle's area, and we know that A-T is some positive amount. If we inscribe a square within our circle and continuously bisect its side, we arrive at a regular polygon inscribed within the circle. That is, if we bisect a square's sides, we obtain an octagon; if we bisect the octagon's sides, we obtain a 16-gon; and so on, until we obtain a regular n-gon whose area is different than the quantity A-T (see Figure 2).

Figure 2

Then, A-Area(inscribed n-gon)<A-T, and by subtracting A from both sides, we obtain  -Area(inscribed n-gon)<-T, which simplifies to Area(inscribed n-gon)>T by dividing both sides by -1. Since our n-gon   is an inscribed polygon,  we know its perimeter , which we denote as Q, is less than C,    an its apothem h is less than r. So, Area( inscribed n-gon=1/2hQ<1/2rC=T, which simplifies to Area(inscribed n-gon)<T. Here we reach a contradiction, since we have shown that T< Area (inscribed n-gon) and T> Area (inscribed n-gon). Then, A cannot be greater than T. 

(2) We examine the case where A<T. Then, we assume that the circle's area A is less than T, so  T-A represents the quantity of excess area  over the circle we can circumscribe a regular n-gon about the circle, whose area exceeds the circle by less than T-A (see Figure 3).

Figure 3

So, Area (circumscribed n-gon)-A<T-A, which can be simplified by adding A to both sides to Area (circumscribed n-gon)<T. The circumscribed n-gon has apothem h which equals radius r, and the n-gon's perimeter Q exceeds C. Then: Area (n-gon)=1/2hQ>1/2rC=T, which means that Area (n-gon)>T. Here we reach a contradiction, since we have shown that T>Area(n-gon) and T<Area(n-gon). This is impossible, so A cannot be less than T. Since we have shown that A cannot be greater than T, and A cannot be less than T, it must be the case that A=T, and our proof is complete.

Consequence (how the area formula comes about),  Recall that  C/D=Pi, or C=DPi. Then, C=2Pir, since D=2r. Using this knowledge, we can find the area of a circle, since we know A=T, and T=1/2rC. 

A=T

A=1/2(rC)

A=1/2(r(2Pir))

A=(1/2)(2)(Pi)(r^2)

A=Pi(r^2). 

So, this is how Archimedes' proof of the area of a circle yielded the familiar formula we use today. Pretty cool, huh? 

What's the Point? 

So, if you're still reading, you may be thinking, "What's the point? What does an ancient Greek mathematician have to teach us, anyway? After all, we have calculators and computers to help us now." I would argue that a scholar such as Archimedes has a lot to teach us about mathematics, thinking, and solving problems in general. Whether or not you consider yourself a "math person," Archimedes can still teach us a valuable lesson about trying and failing. Think about the point I just made about our ability to use technology to help us solve problems now. Archimedes didn't have any of these tools,  he only had a compass, a ruler, and his brain to help him to solve problems. Do you think he conjectured the area of a circle after one attempt? No! He kept on thinking about a way he could solve the problem, most likely drawing on the research of his predecessors, such as Euclid. I like to imagine Archimedes surrounded by crumpled diagrams, drawings, and unfinished proofs, on his 450th attempt at finding a way to prove his proposition and sweating with frustration, much like many of us do when given a particularly difficult problem to solve. However, he didn't solve the problem by giving up. He made mistakes, learned from them, and kept trying to solve the problem. 

The point I'm trying to make here is that in order to be a "genius," we have to try and fail. For every 1 brilliant solution, there is probably over 100 mistakes. (Thomas Edison once said something in a similar vein). As thinkers, we must try and fail, learn and eventually succeed. I would argue that some of the greatest lessons we learn in life come from mistakes, and they help us learn quite a bit more than succeeding in the first round ever would. 

So, when thinking about Archimedes, or any other person who you would consider a "genius," remember that you too have the capacity to be a genius, you just have to keep trying- and failing!