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A New Look at Geometry
By Irving Adler
Dover Publications, Inc.Copyright © 1994 Irving Adler
All rights reserved.
One Book and Three Metaphors
The purposes of this book may be expressed in terms of three metaphors.
From the moment he is born a child begins to explore his environment by seeing and hearing it, by moving around in it, and by touching and manipulating things. Out of these explorations there crystallize his first primitive notions of space and the existence of objects in space. Through his daily experience he acquires conceptions of size and shape and distance. When he goes to school he refines these conceptions by absorbing into his experience some of the ideas about space created by inventive minds in the past. At first he studies geometry informally, with emphasis on mensuration. Later, in high school, he gets a brief introduction to Euclidean geometry as a deductive mathematical system. If he goes on to college he learns of new aspects of geometry presented under the titles "Analytic Geometry" and "Calculus." He may hear hints, too, of other mysterious divisions of geometry that only the specialist penetrates: non-Euclidean Geometry, Projective Geometry, Topology, and others. If he is one of the lucky specialists, he discovers that geometry is a many-faceted gem cut and polished from the raw material of our daily experience. One purpose of this book is to permit the reader to share the pleasure of the specialist as he turns this gem in the light and catches the brilliant flashes of color reflected from its facets.
The Valley and the Mountain
Geometry today consists of many subdivisions. There are synthetic geometry, analytic geometry, and differential geometry. There are Euclidean geometry, hyperbolic geometry, and elliptic geometry. There are also metric geometry, affine geometry, projective geometry, and other branches besides. The subdivisions of geometry have been compared to the distinguishable regions within a complex landscape. Most of these regions are in a valley. An explorer who is deep within one region can easily lose sight of the fact that the other regions exist. At a boundary where one region touches another he can see the fact that the regions are related to each other. But seeing the regions pair by pair does not suffice to reveal the pattern of this relationship. There is a path from the valley that leads up the side of a mountain to a clearing at the top. The explorer who reaches this clearing suddenly sees the whole valley laid out before his eyes. From his height at the top of the mountain he can see all the regions of the valley and the pattern that they form. A second purpose of this book is to lead the reader from region to region in the valley, where he can savor the special beauties for which each is famous, and then to take him up to the top of the mountain where he can see the grand design of the valley in all its breathtaking splendor.
The Motion Picture Film
A motion picture theatre tries to interest the passerby in the film that is being shown by putting on display selected still photographs from the film. The passerby, looking at these "stills," sees people in frozen attitudes of action. However, he knows that each of these pictures is but one of many frames on the film; that these frames form a time sequence; and that if he enters the theatre to see the pictures flashed on the screen in quick succession he will see the action and movement by which the story of the film unfolds. The many subdivisions of geometry are like the still photographs of a motion picture film. If we view them in sequence, they, too, tell a story, the story of the evolution of geometry through five thousand years of history. A third purpose of this book is to show the reader the motion picture as well as the "stills," so that he may see the exciting story of geometry evolving.
Neither Fish Nor Fowl
The form of the book, determined by its threefold purpose, is a compromise between exposition and narration. There is much geometry in the book, but the book is not a textbook of geometry. The sequence in which ideas are developed is approximately chronological, but the book is not a history of geometry. Frequently, when we encounter an idea in an ancient setting, we shall view it with hindsight from the modern point of view, in order to see the full range of its implications. The book is organized around a few basic themes: 1) The relationship between physical space and mathematical space, and our changing conceptions of each. 2) The relationship between algebra and geometry, and how this relationship has changed in the course of time. 3) The story of how three separate streams of thought, the theory of parallels, the theory of curved surfaces, and the geometry of position converged to form one integrated whole. 4) The crystallization of the ideas which will permit us to answer the questions, "What is a space?" and "What is a geometry?"CHAPTER 2
Geometry Before Euclid
The Measurement of Physical Objects
Geometry in its earliest form, as developed in ancient Babylonia and Egypt, was concerned with the measurement of physical objects. It dealt with such practical problems as finding the length of a piece of cloth, the area of a field, or the volume of a basket.
There are three basic steps that are involved in making a measurement: 1) selection of a unit; 2) repetition of the unit; 3) counting the number of times that the unit is repeated. For example, to measure the area of a floor, we may choose a particular square tile as unit, and we count the number of such tiles that must be put side by side in order to cover the floor. The first significant results in geometry were short-cuts for carrying out the third step. For example, if a floor is covered by 5 rows of tiles and there are 3 tiles in each row, it is not necessary to count the tiles one by one to find the area of the floor. It suffices to multiply the numbers 3 and 5. This fact was already known to the priests of ancient Babylonia over five thousand years ago. Though they had no algebraic symbolism with which to express it, they were familiar with the formula for the area of a rectangle, A = hb, where h is the length of the height of the rectangle, and b is the length of its base. The Babylonians also knew the analogous formula for the volume of a prism or cylinder, V = hB, where h is the length of the height of the solid, and B is the area of its base.
The Use of Averages
The Babylonians tried to derive from the formula A = hb a more general rule for computing the area of a quadrilateral from the lengths of its four sides. We have no record of the reasoning that they used, but it has been surmised by some historians that it may have taken the following form: "To find the area of aquadrilateral, first replace it by a rectangle with approximately the same area. If the pairs of opposite sides of the quadrilateral have lengths a, a', and b, b' respectively, use the average of a and a' as the height of the rectangle, and use the average of b and b' as the base of the rectangle. Then, using the formula for the area of a rectangle, we get A = [1/2(a + a')][1/2(b + b')], or A = 1/4(a + a')(b + b')." In any case, the latter formula is the one that was used in Babylonia about 3000 B.C. Unfortunately it gives a correct result Unfortunately it gives a correct result only when the quadrilateral is a rectangle.
Similar reasoning may have been used to derive the Babylonian formula for the volume of a basket whose height is h and whose upper and lower bases have areas B and B' respectively. If the volume of the basket is assumed to be equal to that of a cylinder of the same height whose base has an area equal to the average of B and B', then we get the Babylonian formula for the volume of the basket, V = h[1/2(B + B')] = 1/2h(B + B'). Unfortunately, this formula is correct only if B = B'.
The Egyptians who lived one thousand years later were more ingenious and more successful in using the averaging principle. To compute the volume of a frustum of a square pyramid whose height is h and whose bases have edges a and b respectively, they used the correct formula instead of the Babylonian formula V = 1/3h(a2 + ab + b2) It is surmised that they derived their formula by equating the frustum to a prism whose height is h and whose base area is the average of three areas, namely a2, b2, and ab. The third of these areas, ab, is itself a kind of average between a2 and b2, known as their geometric mean. It is the area of a rectangle whose height is a and whose base is b.
The Sides of a Triangle
The Babylonians and the Egyptians who followed them were aware of the fact that the length of the hypotenuse of a right triangle is related to the lengths of the other two sides. The Babylonians computed the hypotenuse c by means of the approximate formula
For the case where a = 4 and b = 3, this formula yields This is not a bad approximation to the value c = 5 obtained from the later Pythagorean formula c2 = a2 + b2.
If the sides a, b and c of a triangle satisfy the Pythagorean formula, then the angle opposite c is a right angle. The Egyptians were aware of at least a special case of this rule, and used it in their technique for constructing a right angle. The surveyors of that time, known as "rope-stretchers," laid out a right angle with the help of a rope that was divided into equal segments by a series of knots. They used the rope to form a triangle whose sides had the ratio 3 to 4 to 5. The angle opposite the longest side was the sought-for right angle.
c = a + b2/2a.
The Concept of Physical Space
When the Babylonians or Egyptians computed a volume, it was always the volume of a particular physical object, such as a basketful of grain or a block of stone. Long experience with such computations led to the emergence of a new idea, that the volume was not a property of the grain as such, or of the stone as such, but of the space occupied by the grain or stone. This space may be thought of as a container that may be filled with any substance whatever. The same space that contains a basketful of grain may be filled with a basketful of sand instead. The same space that is occupied by a block of stone may be occupied by a block of wood instead. On the basis of this idea, geometry becomes the study of the space occupied by physical objects, rather than a study of the particular physical objects themselves. This space, an abstraction from our experience with physical objects, is an aspect of the physical world in which we live, so we refer to it as physical space.
The first person to formulate the abstract idea of physical space was the Greek merchant, mathematician and philosopher Thales of Miletus (640–546 B.C.). During his active days as a merchant, Thales had often traveled to Egypt. There he learned the geometry of the Egyptians, and brought it back to Greece. When he retired he devoted his time to studying and teaching mathematics and philosophy. Thales knew enough about astronomy to be able to predict the solar eclipse of 585 B.C.
Geometry as the study of physical space is a physical science. Therefore there are two distinct ways in which its propositions can be proved. They can be proved empirically, by experiments in which particular configurations are observed and measured. Or they may be proved by showing that they are logical consequences of other propositions that have already been proved. Thales is credited with the first significant use of the latter type of proof. Among the propositions that were discovered and proved by Thales are these:
The base angles of an isosceles triangle are equal.
An angle inscribed in a semicircle is a right angle.
A diameter of a circle bisects the circle.
Numbers and Space
In the measurement of lengths, areas or volumes, numbers are intimately related to space. In this sense we may say that, from its inception, geometry, the study of space, was fused with arithmetic and algebra, the study of numbers. This fusion was first intensified and then undermined by the work of Pythagoras and his followers.
Pythagoras (about 548 B.C. to 495 B.C.) was a pupil of Thales. Like his master, he traveled in Egypt and absorbed the geometric knowledge of the Egyptian priests. In 529 B.C. he settled in Crotona, in southern Italy, where he became a celebrated teacher of mathematics and philosophy. His pupils organized a religious society known as the Order of the Pythagoreans. In the course of their scientific-philosophic-religious speculations, the Pythagoreans made many important discoveries in geometry. They followed the custom of attributing all these discoveries to Pythagoras himself. Consequently we cannot be sure which of these discoveries were made by Pythagoras and which were made by his pupils.
In the thinking of Pythagoras, the fusion of numbers and space took on a peculiar one-sided form. To Pythagoras, whole numbers were not merely an aspect of space. They were the essence of space and of all things in the universe. The number one, from which all other whole numbers could be generated by addition, was the essence of divinity. The even numbers two, four, six, eight, etc., contained the female principle. The odd numbers, three, five, seven, etc., contained the male principle. Five stood for marriage, since it was the result of uniting the first even with the first odd number. Six was the cause of cold, seven of health, and eight of love.
Somewhat less mystical, and more meaningful mathematically, was the linking of numbers with shapes. To Pythagoras, the numbers one, three, six, ten, and so on, obtained by taking the sums 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, etc., are triangular numbers, and the numbers one, four, nine, and so on, obtained by taking the products 1 × 1, 2 × 2, 3 × 3, etc., are square numbers. This linking of numbers and shape helps to reveal many interesting number relationships.
The Pythagoreans made many important discoveries in geometry. We shall examine now a few of the discoveries that play significant roles in the later evolution of geometric ideas.
The Pythagorean Theorem
The name "Pythagorean Theorem" is reserved for the rule that describes how the lengths of the sides of a right triangle are related. In contemporary textbooks of geometry the rule is usually given as, "The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs." If the legs of the right triangle have lengths a and b respectively, and the hypotenuse has length c, then the rule is expressed by the equation a2 + b2 = c2. The rule did not take this algebraic form for Pythagoras, because algebra had not yet been invented. For Pythagoras the rule was a purely geometric one concerning the areas of squares, and was expressed in this form: "The square on the hypotenuse of a right triangle is equal to the sum of the squares on the legs." The squares referred to are shown in the next diagram. There are many ways of proving this theorem. In the book, The Pythagorean Proposition, by Elisha S. Loomis, 256 different proofs are given! Diagrams I, II and III contain the essence of one proof that goes directly to the heart of the geometric content of the theorem. In diagram I we see a right triangle and the square on each leg. In diagram II, three more triangles have been adjoined to the figure to fill out a large square whose side has length a + b. If we denote by T the area of the triangle, then the area of this square, as shown in diagram
II, is a2 + b2 + 4T. In diagram III we see another way of dissecting this large square into parts. A triangle congruent to the original one is cut off at each corner of the square. The remaining piece is clearly a square whose side has length c. The area of the large square, as shown in diagram III, is c2 + 4T. Equating these two expressions for the same area, we get the equation a2 + b2 + 4T = c2 + 4T. If we remove the four triangles in each of the diagrams II and III, the areas that remain are equal. That is, a2 + b2 = c2.
Excerpted from A New Look at Geometry by Irving Adler. Copyright © 1994 Irving Adler. Excerpted by permission of Dover Publications, Inc..
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