# The geometrical problems. Bryson An ancient Greek mathematician

The method of exhaustion

Creation

The method of exhaustion was a technique

used from ancient Greek mathematicians to solve results that are now solved

with the use of limits. It amounts to an early form of integral calculus and it

was created with the purpose of finding the area of a shape by engraving inside

it a series of polygons whose areas unite to the area of the containing shape.

The difference in area between the nth etched polygon and that of the covering

shape will become smaller as n becomes large, if we execute the sequence with

the correct way. While the space between the incised polygon and the involved

shape becomes extremely small, then the possible values for the area of the covering

shape are methodically “exhausted” by the lower bound polygonal areas

consecutively recognized by the sequence members. The idea of the method of

exhaustion was firstly originated with Antiphon of Athens in the 5th

century BC. The method of exhaustion is considered as a forerunner to the

methods of modern calculus. Between the period of 17th and 19th century, the

development of analytical geometry and rigorous integral calculus (more

specifically in the sector of the limit definition) classified the method of

exhaustion so that it is no longer used today in order to solve geometrical

problems.

Bryson

An

ancient Greek mathematician and sophist named Bryson of Heraclea that was born

around 450 BCE, was the first to engrave a polygon inside a circle, discover

the polygon’s area, twofold the number of sides of the polygon, and repeat the

process, leading to a lower bound approximation for the area of circle. Later

on, Bryson used the same process in pursuance of polygons circumscribing a

circle, resulting in a higher certain approximation for the area of a circle.

Bryson after all these calculations was able to almost accurate ? and further

place lower and upper bounds on ?’s real value. Unfortunately, due to the difficulty

of the method, Bryson was only able to compute ? to a few digits. We will

probably never know faithfully who was first to find out that the ratio between

the area of a circle and the area of a square having side length equal to that

of the circle’s radius.

Archimedes

Archimedes,

one of the greatest mathematicians of all time was always working to produce

formulas because we wanted to calculate the areas of regular shapes. For

instance, he wanted to estimate the area of a circle. To achieve that, he designed

a polygon outside the circle and a smaller one inside it. Each time he enclosed

I bigger polygon in the circle from both side approximating the area of the

circle more closely. This is the method of exhaustion and Archimedes was one of

its first exponents of this method. With this method he managed to discover the

area of a parabolic segment, the volume of a paraboloid, the tangent to a

spiral and also a proof that the volume of a sphere is 2/3 the volume of a

circumscribing cylinder. As for the area of a circle, the way Archimedes stated

his proposition was the area equals to the area of a triangle whose height and

base equals to its radius and to its circumference respectively: (1/2)(r*2?r)=2?r^2. But there is something delicate here. We have never seen a reference

similar with this before in Greek mathematics talking about the length of a

curve opposed to the length of a polygon. In the present, the length of a curve

is defined to be a limit. In fact there are curves with infinite length but

Archimedes is restricting them to a countable value. This was a wise decision

done by him because limits were discovered many years later in about 1820.

Until then, Archimedes method seemed to be the best choice for those years.

Convexity

Archimedes

doesn’t need to know much information regarding the length of the curves, since

a circle is a relatively a simple one. His axiom is concerned only with a

restricted class of curved paths which are called convex paths. These kinds of

paths can be described by examining whether something is convex or not. Here’s

one way to distinguish the convex paths from the others. Convex paths bulge out

while the others have dimples.