8.)
Describe the contributions to geometry made by these mathematicians:
Euclid, Archimedes, Mobius, and Pythagoras.

**Euclid
made many contributions to mathematics, many of which are still used
today. For example, Euclid constructed a deductive system of Geometry,
created of many theorems, some of which are still used as important
aspects of modern mathematics. His most famous contribution, however,
originated from his writings--the best known are his expositions, "The
Elements." His writings are made mostly of the discoveries of earlier
mathematicians, such as Hippocrates, Pythagoras, Thales, and Euxodus.
His works began with definitions, "common notions," and axioms
(postulates), followed by the proofs, basing each proof on the information
before it.**

**Archimedes
made his largest contributions to mathematics in the form of geometry.
One of his most famous works is "Measurement of the Circle."
In this, he determined the exact value of pi of being between 3 10/71
and 3 1/7. To determine this he circumscribed and inscribed a circle
with regular polygons having 96 sides. This, however, did require the
proof of 2 fundamental relations of the perimeters and areas of the
inscribed and circumscribed regular polygons. He also proved that the
volume of a sphere is 2/3 the volume of a cylinder that is circumscribed
about the sphere. He also described how to trisect an angle using his
spiral. First, construct circles of radii with a constant successive
difference. These circles should cut the spiral at equal angles. In
order to trisect a particular angle, trisect a radial line segment that
corresponds to the value of the radii at the intersection of the spiral
and the lines that create the angle (with the vertex at the origin).
Finally, construct circles at the trisection points, centering them
about the origin.**

**August
Ferdinand Mobius published almost all of his work in Crelle's Journal,
the very first journal fully devoted to publishing mathematical finds.
In 1827, Mobius's work ***Der barycentrische Calcul, *on analytical
geometry, was a classic of his, including many of his results on projective
and affine geometry. In this work, he introduces homogeneous coordinates
and discusses geometric transformations, particularly projective transformations.
He also introduced a configuration called the Mobius net, playing an
important role in the development of projective geometry. Another of
his introductions is the Mobius inversion formula, introduced in the
paper *Über eine besondere Art von Umkehrung der Reihen, *in
1831. He also is recognized, yet not the founder, of the Mobius strip.
(Listing was the first to discover and describe this object.) One last
thing that I am going to mention is Mobius's publication of the geometric
treatment of statics, which led to the study of systems of lines in
space. Mobius, however, made many other contributions to the study of
mathematics.

**Pythagoras
is considered an important contributor to the foundation of Greek geometry.
One of Pythagoras's most famous discoveries is the Pythagorean Theorem,
which is used to determine the sides of a right triangle (a^2 + b^2
= c^2). Pythagoras developed many theorems and proved them all, with
the help of his followers, known as the Pythagoreans. He is also said
to be the founder of the existence of irrational numbers. Pythagoras
is also credited with the discovery of the fifth regular solid (polyhedron),
the dodecahedron. A regular polyhedron is "a polyhedron that can
be inscribed in or circumscribed about a sphere." This discovery
required the discovery and proof of the regular pentagon. (The Pythagoreans
are credited with the discovery of the fifth regular polyhedron due
to this simple fact.) One final contribution that I will mention is
the use of geometric algebra. Geometric algebra was created by applying
areas to different equations. Geometric algebra was used to solve quadratic
equations as modern algebra is used today. However, its complex nature
prevented it from being utilized to solve more complicated polynomials.**

**Information
and picture of Euclid were found at http://members.tripod.com/Turkler/euclid.html**

**Information
and picture of Archimedes were found at http://www.math.tamu.edu/~don.allen/history/archimed/archimed.html**

**Information
and picture of Mobius were found at **

**http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Mobius.html**

**Information
on Pythagoras was found at http://www.perseus.tufts.edu/GreekScience/Students/Mike/geometry.html**

**Picture
of Pythagoras was found at **

**http://math-www.uni-paderborn.de/%7Erinkens/veranst/elgeo2001/kapitel1/img/Pythagoras_gross.jpg**