More mathematicans and famous problems

Given the length of a square can you find another square whose area will be twice the area as the first? Once options is guess and checking, but that can take a while. Instead there is an algorithm. Say we have a square, labeled PQRS with side length "p":


 Then area(PQRS) = p^2








The area of the square that we want, Sqaure(ABCD) needs to be twice the area as square(PQRS) so
Area(ABCD) = 2* area(PQRS)
                a^2= 2 *(p^2)
===> thus a= p*sqaure root(2)

Thus you have your new square with twice the area as the first one.

On a related note from last post: another mathematician was Plato(429-347). He was a student of Socrates for 8 years. He traveled about Greece and visited Egypt before journeying to southern Italy and Sicily. Plato perused the truth and wanted to master the 'art of argument'. He was the founder of the Academy in Athens and above the entrance of the academy was the phrase 'Let no one ignorant of geometry enter here'. Plato believed that a "student 'ignorant of geometry' would also be ignorant of logic and hence unable to understand philosophy". (pg 41 Katz)

Plato(left) and Aristotle, his student(right)

Portrait of the academy

 

Academy of Athens today

Plato restricted geometry to a compass and a straight edge since he believed any other instrument would lead you astray.

Plato wrote that philosophers and kings, the ideal rulers of a state, should learn mathematics to exercise their mind and "'to practice calculations, not like merchants or shopkeepers for purposes of buying and selling, but with a view to war and to help in the conversion of the soul itself from the world...."(pg 41 Katz)

Another notable mathematician was Hippocrates of Chios. He was one of the first to try 2 of the 3 famous Greek problems: to square a circle and doubling a cube. Hipporates was able to make progress in squaring a circle by being able to square a lune( figures founded by arcs of two circles). To square a lune is to show that "their areas could be shown equal to certain regions bounded by straight lines." (pg 41 Katz). 

Area(Lune BDC) = Area(triangle ABC)

Unanswered question:

1. Is their a practical purpose for squaring a lune? 
2. How does squaring a lune help towards squaring a circle?

Evaluation:

Interesting: 8

quality: 7

complexity:7