HISTORICAL BACKGROUND OF GEOMETRY
Egyptians (c. 2000 - 500 B.C.)
Ancient Egyptians demonstrated a practical knowledge of geometry through surveying and construction projects. The Nile River overflowed its banks every year, and the river banks would have to be re-surveyed. In the Rhind Papyrus, pi is approximated.
The Egyptians did not separate geometry into a separate field of study, but rather combined it with arithmetic to solve practical problems. Most of what historians know about Egyptian geometry comes from two major works, the
Moscow Papyrus , now displayed in Moscow, and the Rhind Papyrus, discovered in 1858 by Henry Rhind. Created around 1700 B.C., these writings contain mathematical and geometric problems and their solutions. These problems mainly dealt with business and administrative pro blems and were used more of as a handbook, rather than a paper that provided proofs or justifications. The Rhind Papyrus contained 85 problems and the Moscow papyrus had 25 problems.
Much of the portion of the Papyruses devoted to geometry dealt with
problems dealing with measurement. The authors did not explain how
the answer was arrived it, just how to solve the problem. Three
particular areas of interest from these works include finding the area of
a circle, the Pythagorean theorem, and measuring volume.
Finding the Area of a Circle
Problems 41-43, 48, and 50 of the Rhind Papyrus dealt with finding the
area of a circle. According to the Egyptians, the area of a circle is the
same as the square that is semi-inscribed in it.
The author reasoned that the area where the square was outside the circle was the same as the area were the circle was outside the square. therefore, to find the area of a circle, just calculate the area of the square since
both areas were the same.
Pythagorean Theorem
A popular misconception is that the Egyptians knew of and used the
Pythagorean theorem. While the mathematicians of the area did know
how to form right angles by setting the two sides of a triangle to 3 and 4
and the long side to 5, they did not understand the theory behind this
method. This theory was the sum of the square of the sides of a triangle
equals the square of the long side of the triangle. The reason for this is
that the historian Moritz Conter (1829-1920) falsely reported in 1882
that the Eqyptians knew of the theory and this belief has stuck with many
up until the present day.
Measuring Volume
While the Egyptians did not know of the Pythagorean theorem, they
remarkably knew a lot about measuring volume. This was particularly
important at this time since knowing the volume of clay pots and other
containers was essential in trading. Problem 42 of the Moscow papryus
accurately showed how to find the volume of a cylindrical silo, stating
that it is the area of the circle of the silo multiplied by its height.
Problems 44-46 accurately showed how to calculate the volume of a
prism.
Most people assume the Egyptians started geometry through their practical surveying and that the 'leisure class' of priests pursued it for fun. A serious deficiency in Egyptian geometry is the lack of clear cut distinction between relationships that are exact and those that are approximate only. They made however the first exact statements about the interrelationship among geometrical figures (no theorems or formal proofs given however).
Examples of Egyptian GeometryEgyptian rule for area of a circle
'' (4.5) = 64
'' = 3.1605
Ahmes said the area of circular field diameter 9 = area of square with side 8.
i.e. Egyptian taking = 3.1/6. But no hint that Ahmes was aware that they were not exactly equal.
Egyptian rule for the circunference of a circle.
area of circle
area of circumscribed sq
___________
=
__________________
circum.of circle
perimeter of circum.sq
Conclusions on Egyptians
The knowledge displayed in Ahmes and Moscow papyri is mostly practical and calculations were the most important thing. The rules of calculation concern specific concrete cases only. The Egyptians were the premier architects and builders of ancient civilizations. The pyramids and the buildings they created were tremendous in size, complex in their design and built to last forever. Three thousand years, or so, is hardly forever, yet their existence for so long does make the point. The Egyptians also laid the foundations of surveying and measuring, a skill made necessary by the annual flooding of the plain alongside the Nile River. Every spring the Nile floodwaters would erase all semblance of the markings laid out to distinguish boundaries. They devised a system of measurement that surpassed any other. Indeed, the term geometry is derived from the Greek “geometria”, measurement of the earth. It is fair to presume that the Egyptians inspired the coining of this term.
Babylonians (c. 2000 - 500 B.C.)
Ancient clay tablets reveal that the Babylonians knew the Pythagorean relationships. One clay tablet reads
4 is the length and 5 the diagonal. What is the breadth? Its size is not known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth.
Babylonians were interested in geometry from point of view of finding numerical approximation to use in mensuration. They didn't distinguish between exact and approximate.
The Babylonians used the Pythagorean Theorem widely.
eg. A reed stands against a wall. If the top slides down 3 units when the lower end slides away nine units, how long is the reed?
The Babylonians did much in the way of astronomical observation which, in turn, led to the development of dividing time into periods and the prediction of astronomical occurrences. All of this is to point out that the Greeks did not invent mathematics. What they did do is begin to formalize and compile much of the valuable work in the field that preceded them.
Greeks (c. 750-250 B.C.)
The Ancient Greeks practiced centuries of experimental geometry like Egypt and Babylonia had, and they absorbed the experimental geometry of both of those cultures. Then they created the first formal mathematics of any kind by organizing geometry with rules of logic. Euclid's (400BC) important geometry book The Elements formed the basis for most of the geometry studied in schools ever since.
The Greeks built upon a solid foundation that was laid by the Egyptians, Babylonians and other ancient civilizations. The Pythagorean Society made mathematics part of a liberal education. The Pythagoreans divided the mathematical subjects into four main parts:
Numbers absolute (arithmetic)
Numbers applied (music)
Magnitudes at rest (geometry)
Magnitudes in motion (astronomy)
This so-called “quadrivium”, in the belief of the Pythagoreans, was what constituted the necessary course of studies for a liberal education.
Geometry is the main focus of this unit. It is a field of study that rests squarely on a foundation of axioms, from the Greek “axioma”, meanings things that are worthy. These axioms are premises held to be so self-evident that one does not need a proof. A certain amount of faith is needed to give one a point of departure, since these axioms cannot, in fact, be proved. The entire system of geometry is built upon these simplest of concepts.
The basic concepts of geometry are point, line, angle and surface, or plane surface. The definitions of these concepts that follow are also discussed in the lesson plan portion of this unit. We shall examine them here for the purpose of identifying those aspects that require a measure of faith in order to be accepted.
POINT
A point is the location of a point in space, yet it does not occupy space. A point has neither length, nor width, nor thickness, and it is indivisible. It is impossible to see a point, yet we use a dot to represent its existence, even though, by definition, there are an infinite number of points contained in that dot. The dot on this “i” is clearly visible. It has length, width and thickness; therefore, it is composed of a infinite number of points.
A point is an abstract concept. Clearly, a certain amount of faith is needed to support its existence. It is similar to the belief in the existence of atoms before one could actually see them.
LINE
If one were to set a point in motion the result would be a line. A line has length, yet it has no width or thickness. It, too, is an abstract concept.
ANGLE
Two lines which emanate from the same point produce an angle. The lines are called the sides of the angle, and the common point is called the vertex of the angle.
SURFACE or PLANE
If one were to move a line at right angles to its own direction the result is a surface, or plane surface. It is often referred to simply as a plane. Tabletops, walls, a pane of glass, or a floor are all examples of what a plane is, with one important distinction—a plane has no edges. A plane extends indefinitely in all directions. It has length and width, yet no thickness. Planes are two dimensional.
GREEK MATHEMATICIANS
Euclid was a Greek mathematician, who lived circa 300 BC, whose chief work, Elements, is a comprehensive treatise on mathematics in 13 volumes on such subjects as plane geometry, proportion in general, the properties of numbers, incommensurable magnitudes, and solid geometry. He probably was educated at Athens by pupils of Plato. He taught geometry in Alexandria and founded a school of mathematics there. The Data, a collection of geometrical theorems; the Phenomena, a description of the heavens; the Optics; the Division of the Scale, a mathematical discussion of music; and several other books have long been attributed to Euclid; most historians believe, however, that some or all of these works (other than the Elements) have been spuriously credited to him. Historians disagree as to the originality of some of his other contributions. Probably the geometrical sections of the Elements were primarily a rearrangement of the works of previous mathematicians such as those of Eudoxus, but Euclid himself is thought to have made several original discoveries in the theory of numbers. Euclid's Elements was used as a text for 2000 years, and even today a modified version of its first few books forms the basis of high school instruction in plane geometry. The first printed edition of Euclid's works was a translation from Arabic to Latin, which appeared at Venice in 1482.
Thales (625?-546?BC),
Thales was a Greek philosopher, born in Miletus, Asia Minor. He was the founder of Greek philosophy, and was considered one of the Seven Wise Men of Greece. Thales became famed for his knowledge of astronomy after predicting the eclipse of the sun that occurred on May 28, 585 BC. He is also said to have introduced geometry in Greece. According to Thales, the original principle of all things is water, from which everything proceeds and into which everything is again resolved. Before Thales, explanations of the universe were mythological, and his concentration on the basic physical substance of the world marks the birth of scientific thought. Thales left no writings; knowledge of him is derived from an account in Aristotle's Metaphysics.
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