The more that I find out about Pythagoras, the more interesting the guy becomes. I know the guy did not have any of your modern computers and no paper to write on. If so, how was he able to accomplish so much? It turns out that computers, books or colleges don't make you smart. You have to be born smart and then work hard to take advantage of what you were born with. That is what Pythagoras did. I have a simple problem that Pythagoras solved thousands of years ago. Do you want to try your hand at solving it? It has to do with music and stringed instruments of the day.
Pythagoras noticed that on all stringed instruments like guitars, the frets seemed to all be spaced similarly. I say similarly because not all guitars type instruments had the same length strings. What Pythagoras noticed was that the 12th fret was located in the middle of the string. It did not matter how long the string was, but the 12th fret was always in the middle. The other frets got farther and farther apart as they progressed up to the "nut". That is the end of the string next to where you adjust the tension.
Pythagoras noticed that, starting at the 12th fret, you could calculate the 11th fret position by multiplying the length of the string from the bridge to the 12th fret by a constant and that constant was the same for all types of guitars. Next, you could find the location of the 10th fret by multiplying this same number by the distance from the bridge to the 11th fret and so on.
From this data, you should be able to determine what that number was. Remember, Pythagoras did not have a calculator. He solved the problem in his head.
Good luck!
PS-----Try this hint. Pythagoras found that with his "number", he could find the position of the frets by either starting with the 12th fret or half the length of the string and multiply as above to determine the length from the bridge to the 11th fret and so on or he could "divide" the length of the total string and thus get the first fret and so on down to the 12th where it turned out the 12th fret was, again, in the middle of the string. I hope that helps. I have some more hints if this is not enough.
PS----I will give you one more hint. What number times itself 12 times is equal to "2"? All you need is simple algebra to figure this one out. Algebra is just simple logic that was well known in Pythagoras' day. Did you ever wonder why the Harp was shaped the way that it is? Same answer!!!
PS--I gave this problem to my banker and he had the answer in about a minute. Here is how he solved the problem. If you put one dollar in the bank, what interest rate do you need to earn to double you money in 12 years. The answer is 1 minus the 12th root of 2. He knew the interest rate, so he just added 1 and had the answer.
