August 5th is the 217th day of the year.. and the date on which the first income tax law went into effect in the US, in 1861. The law was passed to help cover the expenses of the Civil War, but managed to hang on until 1872. The rate was 3% of all income over $800 a year. I’m sure it was not intended to be a quid-pro-quo, but the US Army abolished the practice of flogging on the same day."Give me money and I'll stop hitting you." Speaking of stopping, the first electric stop light was erected on this date in 1917 in Cleveland.
Astronaut Neil Armstrong and Mathematician Niels Henrik Abel were both born on this day, and Carmen Miranda and Marilyn Monroe both died on this date in history, but only one of them was ever portrayed by Al Harmon as a college student. If you don’t know which, you’ll have to pry the information out of him.
Since this is the 217th day of the year, there are 148 days remaining, exactly the number of episodes of the "Fresh Prince of Bel-air" that were ever made, you can watch one a day for the remainder of the year and see them all.... wow, lucky you.
I mentioned on the third (8/3 Calculus and all that) that Archimedes knew the area between the curve y=x2 and the x-axis from x=0 to x=2. I’ve been researching a little about his writing, and can add for you that to figure out the above, he became the first person to evaluate an infinite geometric series(or at least the first to do it in writing and leave it where we could find it).
Archimedes actually was working on the area inside the parabola. What he actually showed was that if you drew a chord to a parabola, the area of the section between the chord and the parabola is 4/3 the area of the largest inscribed triangle with the chord as a base. In the picture I have shown the graph of y = 4-x2 cut by the chord y=x+2 as an example. The triangle has vertices at (-2,0), (1,3), and (-.5, 3.75).
He did this by showing that if you inscribed two more triangles in the sections of the parabola outside two sides of the triangle given, that they would add up to ¼ of the given triangle. And then if you draw four more outside these two, they will add up to 1/16 of the first. He then set out to show that 1 + ¼ + 1/16 + 1/64…. = 4/3. 1800 years before the invention of calculus, he used the idea of a limit to show
1 + 1/3 (1) = 4/3
1 + ¼ + 1/3 (1/4) = 4/3
1 + ¼ + 1/16 + 1/3 (1/16) = 4/3 and extended this to show
1 + ¼ + 1/16 + … 1/ (4n) + 1/3 (1/(4n) = 4/3 and of course, as n goes to infinity,1/4nalso goes to zero, leaving the sum 4/3.
Simply incredible....
