Cultural Essay III
Nick Boline
9/21/14
Help Received: Textbook,The following website consulted for clarification on the proof of the
Lune (and the image is from here): http://en.wikipedia.org/wiki/Lune_of_Hippocrates
A major problem the Greeks were trying to solve was squaring the circle, or finding a way to
prove that a given shape constructed with straightedge and compass had the same area as the
corresponding circle. Although this proved impossible, Hippocrates was the first to
geometrically prove the area of a curved shape was equal to a corresponding triangle. The shape,
called a lune, is bound by an arc equal to a quarter circle and an arc equal to a semicircle half its
size.
In this picture, the lune AEBFA has the same area as the triangle ABO. Here is the proof:
Since segment AB is the leg of the isosceles right triangle, the diameter of the circle for which
ABCOA is the semicircle is √2AB. From this we find that quarter circle AFBOA and semicircle
AEBDA have the same area. Since the lune and the triangle ABO are equal area subtracted from
equal area, they must have the same area. Behold, you will find this to be true.
This particular problem has helped me frame the types of problems which the Greek
mathematicians were working with during their earlier centuries. The Lune of Hippocrates is
significant because it is the first recorded example of the area of a curved figure being proven
geometrically. This represents a major breakthrough in geometry and mathematics in general.
And although finding the area of curved figures has very practical applications, to me, this seems
like one of the first steps in the development of mathematical interest into more theoretical areas
of math. It also shows the more geometric approach the Greeks preferred to use in their own
mathematics, which could be a result of some aspect of their culture.
Narcissistic numbers are numbers which are equal to the sum of each digit raised to the power
of the number of digits. Example: 153 = 13 + 53 + 33
Vampire numbers are number with n digits that can be formed as a result of multiplying two
numbers of n/2 digits. These multiplicands must be numbers contained in the vampire number.
For example, 1260 is a vampire number because 21 x 60 = 1260.