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Transcript
```Name____________________________________________
Date_______
A BRIEF HISTORY OF GEOMETRY.
Geometry began with a practical need to measure shapes. The word geometry means to “measure
the earth” and is the science of shape and size of things. It is believed that geometry first became
important when an Egyptian pharaoh wanted to tax farmers who raised crops along the Nile
River. To compute the correct amount of tax the pharaoh’s agents had to be able to measure the
amount of land being cultivated.
Around 2900 BC the first Egyptian pyramid was constructed. Knowledge of geometry was
essential for building pyramids, which consisted of a square base and triangular faces. The
earliest record of a formula for calculating the area of a triangle dates back to 2000 BC. The
Egyptians (5000–500 BC) and the Babylonians (4000–500 BC) developed practical geometry to
solve everyday problems, but there is no evidence that they logically deduced geometric facts
from basic principles1.
It was the early Greeks (600 BC–400 AD) that developed the principles of modern geometry
beginning with Thales of Miletus (624–547 BC). Thales is credited with bringing the science of
geometry from Egypt to Greece. Thales studied similar triangles and wrote the proof that
corresponding sides of similar triangles are in proportion.
The next great Greek geometer was Pythagoras (569–475 BC). Pythagoras is regarded as the first
pure mathematician to logically deduce geometric facts from basic principles. Pythagoras
founded a brotherhood called the Pythagoreans, who pursued knowledge in mathematics,
science, and philosophy. Some people regard the Pythagorean School as the birthplace of reason
and logical thought. The most famous and useful contribution of the Pythagoreans was the
Pythagorean Theorem.
Euclid of Alexandria (325–265 BC) was one of the greatest of all the Greek geometers and is
considered by many to be the “father of modern geometry”. Euclid is best known for his 13-book
treatise The Elements. The Elements is one of the most important works in history and had a
profound impact on the development of Western civilization.
Euclid began The Elements with just a few basics, 23 definitions, 5 postulates, and 5 common
notions or general axioms. An axiom is a statement that is accepted as true. From these basics, he
proved his first proposition. Once proof was established for his first proposition, it could then be
used as part of the proof of a second proposition, then a third, and on it went. This process is
known as the axiomatic approach. Euclid’s Elements form the basis of the modern geometry that
is still taught in schools today.
Archimedes of Syracuse (287–212 BC) is regarded as the greatest of the Greek mathematicians
and was also the inventor of many mechanical devices including the screw, the pulley, and the
lever. The Archimedean screw – a device for raising water from a low level to a higher one – is
an invention that is still in use today. Archimedes works include his treatise Measurement of a
1
Whether or not early Egyptians or Babylonians developed geometric theories before the Greeks is still disputed in
the math community.
Circle, which was an analysis of circular area, and his masterpiece On the Sphere and the
Cylinder in which he determined the volumes and surface areas of spheres and cylinders.
There were no major developments in geometry until the appearance of Rene Descartes (1596–
1650). In his famous treatise Discourse on the Method of Rightly Conducting the Reason in the
Search for Truth in the Sciences, Descartes combined algebra and geometry to create analytic
geometry. Analytic geometry, also known as coordinate geometry, involves placing a geometric
figure into a coordinate system to illustrate proofs and to obtain information using algebraic
equations.
The next great development in geometry came with the development of non-Euclidean geometry.
Carl Friedrich Gauss (1777–1855) who along with Archimedes and Newton is considered to be
one of the three greatest mathematicians of all time, invented non-Euclidian geometry prior to
the independent work of Janos Bolyai (1802–1860) and Nikolai Lobachevsky (1792-1856). NonEuclidian geometry generally refers to any geometry not based on the postulates of Euclid,
including geometries for which the parallel postulate is not satisfied. The parallel postulate states
that through a given point not on a line, there is one and only one line parallel to that line. NonEuclidian geometry provides the mathematical foundation for Einstein’s Theory of Relativity.
The most recent development in geometry is fractal geometry. Fractal geometry was developed
and popularized by Benoit Mandelbrot in his 1982 book The Fractal Geometry of Nature. A
fractal is a geometric shape, which is self-similar (invariance under a change of scale) and has
fractional (fractal) dimensions. Similar to chaos theory, which is the study of non-linear systems;
fractals are highly sensitive to initial conditions where a small change in the initial conditions of
a system can lead to dramatically different outputs for that system.
--1. Of the great geometers noted in this history (besides Euclid), who do you feel has made
the greatest contribution to geometry?
2. How would you defend your belief to someone who claimed Euclid to be the most
influential?
```
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