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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 principles.
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. The theory states that the sum of the squares of the legs
of a right triangle equals the square of the hypotenuse.
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 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 nonEuclidean 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 (17921856). Non-Euclidian 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. Non-Euclidian 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.