A Brief History of Greek Geometry

A Little Background:

The word geometry has its roots in the Greek work geometrein, which means “earth measuring”. Before the time of recorded history, geometry originated out of practical necessity; it was the science of measuring land. Many ancient civilizations (Babylonian, Hindu, Chinese, and Egyptian) possessed geometric information. The first geometrical considerations “had their origin in simple observations stemming from human ability to recognize physical form and to compare shapes and sizes” (Historical Topics, 165). There were many circumstances in which primitive people were forced to take on geometric topics, although it may not have been recognized as such. For instance, man had to learn with situations involving distance, bounding their land, and constructing walls and homes. These types of situations were directly related to the geometric concepts of vertical, parallel, and perpendicular.

The geometry of the ancient days was actually just a collection of rule-of-thumb procedures, which were found through experimentation, observation of analogies, guessing, and sometimes even intuition. Basically, geometry in the ancient days allowed for approximate answers, which were usually sufficient for practical purposes. For example, the Babylonians took p to be equal to 3. It is said that the Babylonians were more advanced than the Egyptians in arithmetic and algebra. They even knew the Pythagorean theorem long before Pythagoras was even born. The Babylonians had an algebraic influence on Greek mathematics.

Egyptian geometry was not a science in the way the Greeks viewed geometry. It was more of a grab bag for rules for calculation without any motivation or justification. Sometimes they guessed correctly, but other times they did not. One of their greatest accomplishments was finding the correct formula for the volume of a frustum of a square pyramid. However, they thought that the formula that they had for the area of a rectangle could be applied to any quadrilateral.

Primitive people could not escape geometry in the same way that we cannot escape it today. The concept of the curve was found in flowers and the sun, a parabola was represented by tossing an object, and spider webs posed an excellent example of regular polygons. Symmetry could be seen in many living objects, including man, and the idea of volume had to be addressed when constructing a device to hold water. Historical Topics for the Mathematics Classroom calls this type of geometry “subconscious geometry”. This is the type of geometry that very young children experience as they begin to play with objects. This type of geometry involves concrete objects.

Still before the time of recorded history, man began to consider situations that were more hypothetical. They were able to take the knowledge they had learned from observation of concrete objects and come up with general algorithms and procedures to be used in particular cases. This is what Historical Topics for the Mathematics Classroom refers to as “scientific geometry” (166-7). Procedures such as trial and error, induction, and rule-of thumb were being used to discover. This was mainly the geometry of the Babylonians and the Egyptians. Although there is no evidence that they were able to deductively reason geometric facts from basic principles, it is thought that they paved the way for Greek geometry. Geometry remained this way (“scientific”) until the Greek period.

A Bit of Greek Geometry (600 BC – 400 AD):

To see a chronological outline of the work of Greek geometers, click here. (An entire mathematical chronology can be found by visiting: www-groups.dcs.st-and.ac.uk/~history/Chronology/full.html.)

The Greeks worked to transform geometry into something much different than the “scientific geometry” of the people that worked before them. “The Greeks insisted that geometric fact mush be established, … , by deductive reasoning; …” (Historical Topics, 171). They believed that geometrical truth would be found by studying rather than experimenting. They transformed the former “scientific geometry” into a more “systematic geometry”.

Keep in mind that there exist virtually no first-hand sources of early Greek geometry. Hence, the following is based on manuscripts written hundreds of years after this early Greek geometry had been developed. According to these manuscripts, Thales of Miletus was the one who began early Greek geometry in the sixth century B.C. He is noted as one of the first known to indulge himself in deductive methods in geometry. His credited elementary geometrical findings resulted from logical reasoning rather than intuition and experiment. He insisted that geometric statements be established by deductive reasoning rather than by trial and error. He was familiar with the computations recorded from Egyptian and Babylonian mathematics, and he developed his logical geometry by determining which results were correct.

The next mentioned great Greek geometer is one who quite possibly studied under Thales of Miletus. This geometer is Pythagoras, who founded the Pythagorean school, which was “committed to the study of philosophy, mathematics, and natural science” (Historical Topics, 172). In the area of geometry, the members of this school developed the properties of parallel to prove that the sum of any angles of a triangle is equal to two right angles. They also worked with proportion to study similar figures. The deductive side of geometry was further developed during this time. We all think of the Pythagorean Theorem when we think of Pythagoras, however it is important to note that this theorem was used (although it may not have been proved) before his time.

As an interesting side note, Pythagoras was regarded as a religious prophet by his contemporaries. He preached the immortality of the soul and reincarnation, and he even organized a brotherhood of believers. This brotherhood had initiation rites, they were vegetarian, and they shared all property. They did, however, differ from other religious groups in one major way. They believed that elevation of the soul and union with God was achieved through the study of music and mathematics.

Hippocrates of Chios was one of these students at the Pythagorean school. It is suggested that he was the first to attempt “a logical presentation of geometry in the form of a single chain of propositions …” (Historical Topics, 172). He is credited to writing the first “Elements of Geometry” where he included geometric solutions to quadratic equations and some of the first methods of integration. He studied the problem of squaring a circle and squaring a lune. He also was the first to show that the ratio of the areas of two circles equals the ratio of the squares of the circles’ radii (www.geometryalgorithms.com/history.htm).

Although Plato did not make any major mathematical discoveries himself, he did emphasize the idea of proof. He insisted on accuracy, which helped pave the way for Euclid. It is correct to say that almost every significant geometrical development can be traced back to three outstanding Greek geometers: Euclid, Archimedes, and Apollonis. Euclid collected the theorems of Pythagoras, Hippocrates, and others into a work called “The Elements”. (www.geometryalgorithms.com/history.htm). Euclid is the most widely read author in the history of mankind. The teaching of geometry has been dominated by Euclid’s approach to the subject. In fact, Euclid’s axiomatic method is the prototype for all “pure mathematics”. By “pure”, it is meant that all statements can be verified through reasoning of demonstrations; no physical experiments are necessary.

Typically, the next mentioned Greek mathematician is regarded as the greatest Greek mathematician by geometryalgorithms.com. His name was Archimedes of Syracuse. He had many mathematical accomplishments as well as being the inventor of the screw, the pulley, the lever, and other mechanical devices. He perfected integration using the method of exhaustion discovered my Eudoxus, and he was able to find the areas and volumes of many objects. Inscribed on his tomb was the result he found that the volume of a sphere is two-thirds the volume of its circumscribed cylinder. (http://geometryalgorithms.com/history.htm)

Apollonius was an astronomer who had his mathematical bid to fame in his work entitled Conic Sections. It is this great Greek geometer who provided us with the terms “ellipse,” “parabola,” and “hyperbola.” He is also accredited with showing how to construct a circle which is tangent to three objects. His approximation of π was even closer than that of Archimedes. (http://geometryalgorithms.com/history.htm)

Last, but certainly not least, Hypatia of Alexandria was the first woman to substantially contribute to mathematics. She studied under her father and assisted him in writing a new version of Euclid’s Elements. She also wrote commentaries on other great Greek geometer’s works. She was the “first woman in history recognized as a professional geometer and mathematician” (http://geometryalgorithms.com/history.htm).

References:

Greenberg, Marvin J. Euclidean and Non-Euclidean Geometries:

Development and History. 3rd ed. New York: W.H. Freeman and Company,
1993. 6-19.

Historical Topics for the Mathematics Classroom. Washington D.C.:

National Council of Teachers of Mathematics, 1969.

http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Thales.html (March 2005)

http://geometryalgorithms.com/history.htm (March 2005)

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