The geometrical problems. Bryson An ancient Greek mathematician

           The method of exhaustion

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   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.