Mathematics for the Nonmathematician

CHAPTER 6 THE NATURE AND USES OF EUCLIDEAN GEOMETRY

Circles to square and cubes to double

Would give a man excessive trouble.

MATTHEW PRIOR

6–1 THE BEGINNINGS OF GEOMETRY

Just as the study of numbers and its extensions to algebra arose out of the very practical problems of keeping track of property, trading, taxation, and the like, so did the study of geometry develop from the desire to measure the area of pieces of land (or geodesy in general), to determine the volumes of granaries, and to calculate the dimensions and amount of material needed for various structures.

The physical origin of the basic figures of geometry is evident. Not only the common figures of geometry but the simple relationships, such as perpendicularity, parallelism, congruence, and similarity, derive from ordinary experiences. A tree grows perpendicular to the ground, and the walls of a house are deliberately set upright so that there will be no tendency to fall. The banks of a river are parallel. A builder constructing a row of houses according to the same plan wishes them to have the same size and shape, that is, to be congruent. A workman or machine producing many pieces of a particular item makes them congruent. Models of real objects are often similar to the object represented, especially if the model is to be used as a guide to the construction of the object.

The science of geometry, indeed, the science of mathematics, was founded by the Greeks of the classical period. We have already described the major steps: the recognition that there are abstract concepts or ideas such as point, line, triangle, and the like, which are distinct from physical objects, the adoption of axioms which contained the surest knowledge about these abstractions man can obtain, and the decision to prove deductively any other facts about these concepts. The Greeks converted the disconnected, empirical, limited geometrical facts of the Egyptians and Babylonians into a vast, systematic, and thoroughly deductive structure.

Although the Greeks also studied the properties of numbers, they favored geometry. The reasons are pertinent. First of all, the Greeks liked exact thinking, and found that this faculty was more readily applied to geometry. Possible theorems are rather easily gleaned from the visualization of geometrical configurations. The neat correspondence between deductively established conclusions and intuitive understanding further increases this appeal of geometry. That one can draw pictures to represent what one is thinking about in geometry has its drawbacks. One is prone to confuse the abstract concept with the picture and to accept unconsciously properties of the picture. Of course, the idea of a triangle must be distinguished from the triangle drawn in chalk or pencil, and no properties of the picture may be used unless they are contained in the axioms or in some previously proved theorem. The Greeks were careful to make this distinction.

Secondly, the Greek philosophers who founded mathematics were intrigued with the design and structure of the universe, and they studied the heavens, certainly the most impressive spectacle in nature, to fathom the design. The shapes and paths of the heavenly bodies and the over-all plan of the solar system were of interest. On the other hand, they hardly saw any value in the ability to describe the exact locations of the moon, sun, and planets and to predict their precise locations at a given time, information of importance in calendar reckoning and in navigation.

Thirdly, since commerce and daily business were handled in large part by slaves, and were in any case in low regard, the study of numbers, which served such purposes, was subordinated. Why worry about the uses of numbers for measurement and trade if one does not measure or trade? One does not need the dimensions of even one rectangle to speculate about the properties of all rectangles.

The Greek philosophers emphasized an aspect of reality which is today, at least in scientific circles, neglected. To the Greeks of the classical period the reality of the universe consisted of the forms which matter possessed. Matter as such was formless and therefore meaningless. But an object in the shape of a triangle was significant by the very fact that it was triangular.

Finally, there were purely mathematical grounds for the Greek emphasis on geometry. The Greeks were the first to recognize that quantities such as , etc., are neither whole numbers nor fractions, but they failed to recognize that these were new types of numbers, and that one could reason with them. To handle all types of quantities, they conceived the idea of treating them as line segments. As line segments, the hypotenuse of a right triangle (Fig. 4–2) and the arms have the same character, despite the fact that if the arms are each 1 unit long, the hypotenuse has the irrational length . To execute their plan of treating all quantities geometrically, the Greeks converted the algebraic processes developed in Egypt and Babylonia into geometrical ones. We could illustrate how the Greeks solved equations geometrically, but their methods are no longer favored. For science and engineering, the knowledge that a certain line segment solves an equation is not nearly so useful as a numerical answer which can be calculated to as many decimal places as needed. But the classical Greeks, who regarded exact reasoning as paramount in importance and who deprecated practical applications, found the solution of their difficulty in geometry and were content with this solution. Geometry remained the basis for all exact mathematical reasoning until the seventeenth century, when the needs of science forced the shift to number and algebra and the ultimate recognition that these could be built up as logically as geometry. In the intervening centuries arithmetic and algebra were regarded as practical disciplines.

Of course, the Greek conversion of exact mathematics to geometry was, from our present viewpoint, a backward step. Not only are the geometrical methods of performing algebraic processes insufficient for science, engineering, commerce, and industry, but they are by comparison clumsy and lengthy. Moreover, because Greek geometry was so complete and so admirable, mathematicians following in the Greeks’ footsteps continued to think that exact mathematics must be geometrical. As a consequence, the development of algebra was unnecessarily delayed.

6–2 THE CONTENT OF EUCLIDEAN GEOMETRY

The major book on geometry of the classical Greek era is Euclid’s Elements, a work on plane and solid geometry. Written about 300 B.C., it contains the best results produced by dozens of fine mathematicians during the period from 600 to 300 B.C. The work of Thales, the Pythagoreans, Hippias, Hippocrates, Eudoxus, members of Plato’s Academy, and many others furnished the material which Euclid organized. His text was not the first to be written, but unfortunately we do not have copies of the earlier ones. It is quite certain that the particular axioms one finds in the Elements, the arrangement of the theorems, and many of the proofs are all due to Euclid. The geometry texts used in high schools today in essence reproduce Euclid’s work, although these contemporary versions usually contain only a small part of the 467 theorems and many corollaries found in the Elements. Euclid’s version is so marvelously knit together that most readers are amazed to see so many profound theorems deduced from the few self-evident axioms.

Though the reader may already be familiar with the basic theorems of Euclidean geometry, we shall take a few moments to review some features of the subject and the nature of the accomplishment. We might note first the structure of Euclid’s Elements. He begins with some definitions of the basic concepts: point, line, circle, triangle, quadrilateral, and the like. Although modern mathematicians would make some critical comments about these definitions, we shall not discuss them at present. (See Chapter 20.)

Euclid then states ten axioms on which all subsequent reasoning is based. We shall note these merely to see that they do indeed describe apparently unquestionable properties of geometric figures. The first five axioms are:

AXIOM 1. Two points determine a unique straight line.

AXIOM 2. A straight line extends indefinitely far in either direction.

AXIOM 3. A circle may be drawn with any given center and any given radius.

AXIOM 4. All right angles are equal.

Fig. 6–1.
The parallel axiom.

AXIOM 5. Given a line l (Fig. 6–1) and a point P not on that line, there exists in the plane of P and l and through P one and only one line m, which does not meet the given line l.

In a separate definition Euclid defines parallel lines to be any two lines in the same plane which do not meet, that is, do not have any points in common. Thus, Axiom 5 asserts the existence of parallel lines.

The remaining five axioms are:

AXIOM 6. Things equal to the same thing are equal to each other.

AXIOM 7. If equals be added to equals, the sums are equal.

AXIOM 8. If equals be subtracted from equals, the remainders are equal.

AXIOM 9. Figures which can be made to coincide are equal (congruent).

AXIOM 10. The whole is greater than any part.

The formulations of these axioms are not quite the same as those prescribed by Euclid. Axiom 5 is, in fact, different from Euclid’s, but is stated here in the form which is most likely to be familiar to the reader. The differences between Euclid’s versions and those introduced by later mathematicians are not important for our present purposes, and so we shall not take time now to note them. (See Chapter 20.)

After stating his axioms, Euclid proceeded to prove theorems. Many of these theorems are indeed simple to prove and obviously true of the geometrical figures involved. But Euclid’s purpose in proving them was to play safe. As we shall see in later chapters, many a conclusion seems obvious but is false. Of course, the major proofs are those which establish conclusions that are not at all obvious and, in some cases, even come as a surprise.

Partly to refresh our memories about some theorems of Euclidean geometry and partly to note once again the deductive procedure of mathematics, let us review one or two proofs. A basic theorem of Euclidean geometry asserts the following:

THEOREM 1. An exterior angle of a triangle is greater than either remote interior angle of the triangle.

Fig. 6–2.
An exterior angle of a triangle is greater than either remote interior angle.

Before proving this theorem, let us be clear about what it says. Angle D, in Fig. 6–2, is called an exterior angle of triangle ABC because it is outside the triangle and is formed by one side, BC, and an extension of another side, AC. With respect to angle D, angles A and B are remote interior angles of triangle ABC, whereas angle C is an adjacent interior angle. Hence we have to prove that angle D is larger than angle A and larger than angle B. Let us prove that angle D is larger than angle B.

Fig. 6–3

The problem before us is a tantalizing one because, while it does seem visually obvious that angle D is greater than angle B, there is no apparent method of proof. An idea is needed, and this is supplied by Euclid. He tells us to bisect side BC (Fig. 6–3), to join the mid-point E of BC to A, and to extend AE to the point F, so that AE = EF. He then proves that triangle AEB is congruent to triangle CEF, that is, that the sides and angles of one triangle are equal, respectively, to the sides and angles of the other. This congruence is easy to prove. Euclid had previously proved that vertical angles are equal, and we see from Fig. 6–3 that angles 1 and 2 are vertical angles. Further, the fact that E is the mid-point of BC means that BE = EC. Moreover, we constructed EF to equal AE. Hence, in the two triangles in question, two sides and the included angle of one triangle are equal to two sides and the included angle of the other. But Euclid had previously proved that two triangles are congruent if merely two sides and the included angle of one are equal to two sides and the included angle of the other. Since these facts are true of our triangles, the two triangles must be congruent.

Because triangles AEB and CEF are congruent, angle B of the first triangle equals angle 3 of the second one. We know that angle 3 is the angle to choose in the second triangle as the angle which corresponds to B, because angle B is opposite AE, and angle 3 is opposite the corresponding equal side EF. The proof is practically finished. Angle D is larger than angle 3 because the whole, angle D in our case, is greater than the part, angle 3. Hence angle D is also greater than angle B because angle B has the same size as angle 3.

We have now proved a major theorem, and we should see that a series of simple deductive arguments leads to an indubitable result.

And now let us prove another, equally important theorem which will exhibit one or two other features of Euclid’s work:

THEOREM 2. If two lines are cut by a transversal so as to make alternate interior angles equal, then the lines are parallel.

Fig. 6–4

Again let us see what the theorem means before we consider its proof. In Fig. 6–4, AB and CD are two lines cut by the transversal EF. The angles 1 and 2 are called alternate interior angles, and we are told that they are equal. The theorem asserts that, under this condition, AB must be parallel to CD. As in the case of the preceding theorem, the assertion is seemingly correct, and yet the method of proof is by no means apparent.

Here Euclid uses what is usually called the indirect method of proof; that is, he supposes that AB is not parallel to CD. Two lines that are not parallel must, by definition, meet somewhere. Thus AB and CD meet, let us say, in the point G. But now EG, GF, and FE form a triangle. Angle 2 is an exterior angle of this triangle and angle 1 is a remote interior angle. Since we have the theorem that in any triangle an exterior angle is greater than either remote interior angle, it follows that angle 2 must be greater than angle 1. But, in the above figure, we were given as fact that angle 2 equals angle 1. We have arrived at a contradiction which, if we did not make any mistakes in reasoning, has only one explanation: somewhere we introduced a false premise. We find that the only questionable fact is the assumption that AB is not parallel to CD. But there are only two possibilities, namely, that AB is parallel to CD or that it is not parallel to CD. Since the latter supposition led to a contradiction, it must be that AB is parallel to CD. Thus the theorem is proved.

We should be sure to note that the indirect method of proof is a deductive argument. The essence of the argument is that if AB is not parallel to CD, then angle 2 must be greater than angle 1. But angle 2 is not greater than angle 1. Hence it is not true that AB is not parallel to CD. But AB is or is not parallel to CD. If nonparallelism is not true then parallelism must hold.

Though we shall use a few other theorems of Euclidean geometry in subsequent work, we shall not present their proofs. We are now reasonably familiar with the nature of proof in geometry, and so we shall merely state the theorems when we wish to use them.

Perhaps one other point about the contents of the Elements warrants attention. A superficial survey of the many different theorems may leave one with the impression that the Greek geometers proved what they could and produced merely a mélange. But there are broad themes in Euclidean geometry, and these are pursued systematically. The first major theme is the study of conditions under which geometric figures must be congruent. This is a highly practical subject. Suppose, for example, that a surveyor has two triangular pieces of land and wishes to show that they are equal or congruent. Must he measure all the sides and all the angles of the first piece and show that they are of the same size, respectively, as the sides and angles of the second piece? Not at all! There are several Euclidean theorems which can aid the surveyor. If he can show, for example, that two angles and the included side of the first triangle equal, respectively, the two angles and the included side of the second one, then Euclid’s theorem tells him that the triangular pieces of land must be equal.

A second major theme in Euclid’s work is the similarity of figures, that is, figures with the same shape. We have already mentioned that models of houses, ships, and other large structures are often built to assist in planning. One may wish to know what conditions will guarantee the similarity of the model and the actual structure. Let us suppose that the model or some part of it is triangular in shape. One of Euclid’s theorems tells us that if the corresponding sides of two triangles have the same ratio, then the two triangles will be similar. Thus, if the model is constructed so that each side of the model is of the corresponding side of the actual structure, we know that the model will be similar to the structure. This similarity is useful because, by definition, two triangles are similar if the angles of one equal the corresponding angles of the other. Hence, an engineer can measure the angles of the model and know precisely what the angles of the actual structure will be.

Suppose that two figures are neither congruent nor similar. Could they have some other significant property in common? One answer, clearly, is area. And so Euclid considers conditions under which two figures may have the same area, or, in Euclid’s language, be equivalent.

Fig. 6–5.
The five regular polyhedra.

There are many other themes in Euclid, such as interesting properties of circles, quadrilaterals, and regular polygons. He also considers the common solid figures such as pyramids, prisms, spheres, cylinders, and cones. Finally, Euclid devotes considerable space to a class of figures which all Greeks favored, the regular polyhedra (Fig. 6–5).

EXERCISES

1. What essential fact distinguishes axioms from theorems?

2. Why were the Greeks willing to accept the statements 1 through 10 above as axioms?

3. Use the indirect method of proof to show that if two angles of a triangle are equal, then the opposite sides are equal. [Suggestion: Suppose that angle A (Fig. 6–6) equals angle C, but that BC is greater than BA. Lay off BC’ = BA and draw AC’. Use the theorem that the base angles of an isoceles triangle are equal and Theorem 1 above.]

Fig. 6–6

Fig. 6–7

4. Use the indirect method of proof to show that if two lines are parallel, alternate interior angles must be equal. [Suggestion: Suppose angle 1 in Fig. 6–7 is greater than angle 2. Then draw GH so that angle 1′ equals angle 2. Now use Theorem 2 and Axiom 5.]

5. In Section 3–7, we have briefly outlined the proof that the sum of the angles of a triangle is 180°. Write out the full proof.

6. Under what conditions would two parallelograms be congruent?

7. What conditions would ensure the similarity of two rectangles?

8. A right triangle has an arm 1 mi long and a hypotenuse 1 mi plus 1 ft long. How long is the other arm? Before you apply mathematics, use your imagination to estimate the answer. To work out the problem, use the Pythagorean theorem which says that the square of the hypotenuse equals the sum of the squares of the arms.

9. A farmer is offered two triangular pieces of land. The dimensions are 25, 30, and 40 ft and 75, 90, and 120 ft, respectively. Since the dimensions of the second one are 3 times the dimensions of the first, the two triangles are similar. The price of the larger piece is 5 times the price of the smaller one. Use intuition, measurement, or mathematical proof to decide which is the better buy in the sense of price per square foot.

Fig. 6–8

10. Suppose a roadway is to be built around the earth and each point on the surface of the roadway is to be 1 ft above the surface of the earth (Fig. 6–8). Given that the radius of the earth is 4000 mi or 21,120,000 ft, estimate by how much the length of the roadway would exceed the circumference of the earth. Then use the fact that the circumference of a circle is 2π times the radius and calculate how much longer the roadway would be.

11. Criticize the statement: Euclid assumes that two parallel lines do not meet.

6–3 SOME MUNDANE USES OF EUCLIDEAN GEOMETRY

The creation of Euclidean geometry was motivated by the desire to learn the properties of figures in the world about us. Let us see now whether the knowledge can be applied to the world to good advantage.

Suppose a farmer has 100 feet of fencing at his disposal and he wishes to enclose a rectangular piece of land. Since the perimeter will be 100 feet, the farmer can enclose a piece of land 10 feet by 40 feet, 15 feet by 35 feet, 20 feet by 30 feet, or of still other dimensions, all of which yield a perimeter of 100 feet. The farmer plans to garden in the enclosed plot and therefore wishes the enclosed area to be as large as possible. He notes that the dimensions 10 by 40 would yield an area of 400 square feet; the dimensions 15 by 35 enclose 525 square feet; and the dimensions 20 by 30 enclose 600 square feet. Evidently the area can vary considerably despite the fact that the perimeter in each case is 100 feet. The question then arises, What dimensions would yield the maximum area?

Our first task in seeking to answer this question is to make some reasonable conjecture about these dimensions. We might then be able to prove that the conjecture is correct. Since in the present instance it is easy to play with the numbers involved, let us make a little table of dimensions (always yielding a perimeter of 100 feet) and the corresponding area.

Dimensions, in feet	Area, in square feet
1 by 49	49
5 by 45	225
10 by 40	400
15 by 35	525
20 by 30	600

Study of the table suggests that the more nearly equal the dimensions are, the larger is the area. Hence one might readily conjecture that if the dimensions were equal, that is, if the rectangle were a square, the area would be a maximum.

We can see at once that the dimensions 25 by 25 give an area of 625 square feet, and this area is larger than any of the areas in the table. So far our conjecture is confirmed. However, we could not be sure that some other dimensions, perhaps by , would not do even better. Moreover, even if we could be certain that the square furnishes the largest area among all rectangles with a perimeter of 100 feet, the question would arise whether the square would continue to be the answer for some other perimeter. Hence, let us see whether we can prove the general theorem that of all rectangles with the same perimeter, the square has maximum area.

Figure 6–9 shows the rectangle ABCD. Since this rectangle is not a square, let us erect on the longer side a square which has the same perimeter. Thus, the square EFGD has the same perimeter as ABCD. We now denote equal segments by the same letters. The perimeter of the rectangle is then 2x + 2u + 2y, and the perimeter of the square is 2x + 2v + 2y. Since the two figures have the same perimeter, we have

2x + 2v + 2y = 2x + 2u + 2y.

If we subtract 2x and 2y from both sides of this equation and then divide both sides by 2, we obtain

Moreover, because the square has equal sides,

If we now multiply the left side of equation (2) by the left side of equation (1), and do the same for the right sides, the results must be equal. Hence,

yv = u(x + v),

or, by the distributive axiom,

yv = ux + uv.

Since yv = ux plus an additional area, it must be that yv is greater than ux. Now yv is area B in the figure, and ux is area A. Thus B is greater than A, and so B + C is greater than A + C. But B + C is the area of the square, and A + C is the area of the rectangle. Hence the square has more area than the rectangle.

We have proved that a square has more area than a rectangle of the same perimeter, no matter what this perimeter may be. A little thinking proves in a few minutes what may have taken man hundreds of years to learn through trial and error.

Fig. 6–9.
Of all rectangles of the same perimeter the square has the greatest area.

The result is far more useful than may appear at first sight. Suppose a house is to be built. The major consideration is to have as much floor area or living space as possible. Now the perimeter of the floor determines the number of feet of wall that will be needed and hence the cost of the walls. To obtain the maximum floor area for a given cost of walls, the shape of the floor should be square.

A farmer who seeks the rectangle of maximum area with given perimeter might, after finding the answer to his question, turn to gardening, but a mathematician who obtains such a neat result would not stop there. He might ask next, Suppose we were free to utilize any quadrilateral rather than just rectangles, which one of all quadrilaterals with the same perimeter has maximum area? The answer happens to be a square, though we shall not prove it. The mathematician might then consider the question, Which pentagon of all pentagons with the same perimeter has maximum area? One can show that the answer is the regular pentagon, that is, the pentagon whose sides are all equal and whose angles are all equal. Now the square also has equal sides and equal angles. Hence it would seem that if one compares all polygons of the same perimeter and same number of sides, then the one with equal sides and equal angles, i.e., the regular polygon, should have maximum area. This general result can also be proved.

But now an obvious question comes to the fore. The square has maximum area among all quadrilaterals of the same perimeter. The regular pentagon has maximum area among all pentagons of the same perimeter. Suppose that we compared the regular pentagon with the square of the same perimeter. Which would have more area? The answer, perhaps surprising, is the regular pentagon. And now the conjecture seems reasonable that of two regular polygons with the same perimeter, the one with more sides will have more area. This is so. Where does this result lead? One can form regular polygons of more and more sides, which all have the same perimeter. As the number of sides increases, the area increases. But as the number of sides increases, the regular polygon approaches the circle in shape. Hence the circle should have more area than any regular polygon of the same perimeter. And since the regular polygon has more area than an arbitrary polygon, the circle has more area than any polygon with the same perimeter. This result is a famous theorem.

Now the sphere, among surfaces, is the analogue of the circle among curves. Hence, a reasonable conjecture would be that the spherical surface bounds more volume than any other surface with the same area. This conjecture can be proved. Nature obeys this mathematical theorem. For example, if one blows up a rubber balloon, the balloon assumes a spherical shape. The reason is that the rubber must enclose the volume of air blown into the balloon and the rubber must be stretched. But rubber contracts as much as possible. The spherical figure requires less surface area to contain a given volume of gas than does any other shape. Hence, with the spherical shape, the rubber is stretched as little as possible.

The problem of bounding the greatest possible area with a perimeter of given length has a variation whose solution shows how ingenious mathematical reasoning can be. Suppose that a person has a fixed amount of fencing at his disposal and wishes to enclose as much area as possible along a river front in such a way that no fencing is required along the shore itself. The question now is, What should the shape of the boundary curve be? According to a legend, which may or may not have a factual basis, this problem was solved thousands of years ago by Dido, the founder of the city of Carthage on the Mediterranean coast of Africa. Dido, the daughter of the king of the Phoenician city of Tyre, ran away from home. She took a fancy to this land on the Mediterranean, and made an agreement to pay a definite sum of money for as much land as “could be encompassed by a bull’s hide.” Dido thereupon took a bull’s hide, cut it up into thin long strips, tied the strips together, and used this length to “encompass land.” She chose an area along the shore, because she was smart enough to realize that no hide would be needed along the shore. But there still remained the question of what shape to use for the boundary formed by the hide, that is, for ABC of Fig. 6–10. Dido decided that the most favorable shape was a semicircle, enclosed that shape, and built a city there.

Fig. 6–10

Fig. 6–11

A sequel to this story, which has nothing to do with the mathematics of Dido’s problem, is not without relevance to the history of mathematics. Shortly after she founded Carthage, Aeneas, a refugee from Troy, intent on getting to Italy to found his own city, was blown ashore along with his compatriots. Dido took a fancy to Aeneas also, and did her best to persuade him to remain at Carthage, but despite the best of hospitality, Aeneas could not be diverted from his plan, and soon sailed away. Rejected and scorned, Dido was so despondent that she threw herself on a blazing pyre just as Aeneas sailed out of the harbor. And so an ungrateful and unreceptive man with a rigid mind caused the loss of a potential mathematician. This was the first blow to mathematics which the Romans dealt.

Dido’s fate was a tragic end to a brilliant beginning, for her solution to the geometrical problem described above was correct. The answer is a semicircle. We do not know how Dido found the answer, but it can be obtained very neatly. The way to prove it is by complicating the problem. Suppose that, instead of bounding an area on one side of the seashore, which we idealize as the line AC (Fig. 6–11), we try to solve the problem of enclosing an area on both sides of AC with double the length of hide Dido had for one side, i.e., now we seek to solve the problem by determining the maximum area which can be completely enclosed by a perimeter of given length. The answer to this problem is a circle. If, therefore, we choose a semicircle for arc ABC, it will contain maximum area on one side of the shore. For if there were a more favorable shape than the semicircle, the mirror image in AC of that shape would, together with the original, do better than the circle and yet have the same perimeter as the circle. But this is impossible.

Our last few pages have dealt with problems which grew out of determining the rectangle of maximum area with given perimeter. We can see from the lines of thought pursued how the mathematician can raise one question after another on this same theme of figures with maximum area and given perimeter and will find the answers to these questions. Moreover, many of these answers prove to be applicable to physical problems.

Fig. 6–12.
Eratosthenes’ method of deducing the circumference of the earth.

The first reasonably accurate calculation of the size of the earth was made by a simple application of Euclidean geometry. One of the most learned men of the Alexandrian Greek world, Eratosthenes (275–194 B.C.), a geographer, mathematician, poet, historian, and astronomer, used the following plan. At the summer solstice, the sun shone directly down into a well at Syene (C in Fig. 6–12). As Eratosthenes well appreciated, this meant that the sun was directly overhead. At the same time, at the city of Alexandria, 500 miles north of Syene, the direction of the sun was AS′, whereas the overhead direction was OAD. Now the sun is so far away that the lines AS′ and CS could be taken to be parallel. Eratosthenes measured the angle DAS′ and found it to be 74°. But this angle equals the vertical angle OAE, and the latter and angle AOC are alternate interior angles of parallel lines. Hence angle AOC is also , or , or 1/48 of the entire angle at O. Then arc AC is 1/48 of the entire circumference. Since AC is 500 miles, the entire circumference is 48 · 500 or 24,000 miles.

Strabo, a Greek geographer who lived in the first century B.C., tells us that after Eratosthenes obtained this result, he realized that one might sail from Greece past Spain across the Atlantic Ocean to India. This is, of course, what Columbus attempted. Fortunately or unfortunately, the geographers who lived after Eratosthenes, notably Poseidonius (first century B.C.) and Ptolemy (second century A.D.), gave other results which were interpreted by Columbus (because of some uncertainty about the units of distance used by these early scientists) to mean that the circumference of the earth is 17,000 miles. Had he known the correct value, he might never have undertaken to sail to India because the greater distance might have daunted him.

EXERCISES

1. Suppose that DF (Fig. 6–13) is the course of a railroad, and A and B are two towns. It is desired to build a station somewhere on DF so that the station will be equally distant from A and B. Where should the station be built? One draws the line AB and, at its mid-point, erects the perpendicular CE. The point E on DF is equidistant from A and B. Prove this statement.

Fig. 6–13

Fig. 6–14

Fig. 6–15

2. A pinhole camera is a practical device if a long exposure time is possible. In fact, one of the best pictures of the scene following the explosion of the first atomic bomb was made with a pinhole camera. The principle involves similar triangles. The object AB being photographed (Fig. 6–14) appears on the film inside the box as A′B′. If one draws OD perpendicular to AB, the extension of OD to D′ will be perpendicular to A′B′. Then triangles OAD and OA′D′ are similar. Now suppose the sun, whose radius is AD, is photographed. We know that OD is 93,000,000 mi. Suppose that OD′, the width of the box, is 1 ft. The length A′B′ is readily measured and is found to be 0.009 ft. What is the radius of the sun?

3. A farmer has 400 yd of fencing and wishes to enclose a rectangle of maximum area. What dimensions should he choose?

4. A farmer has p yd of fencing and wishes to enclose a rectangle of maximum area. What dimensions should he choose?

5. A farmer plans to enclose a rectangular piece of land alongside a lake; no fencing is required along the shoreline AD (Fig. 6–15). He has 100 ft of fence and wishes the area of the rectangle to be as large as possible. What dimensions should he choose?

6. Of any two numbers whose sum is 12, the product is greatest for 6 and 6; that is, 6·6 is greater than 5·7, 4·8, , and so forth. Can you explain why this is so? [Suggestion: Think in geometrical terms.]

7. Suppose h is the known height of a mountain, and R is the radius of the earth (Fig. 6–16). How far is it from the top of the mountain to the horizon; that is, how long is x? [Suggestion: Use the fact that the line of sight from the top of the mountain to the horizon is tangent to the circle shown, and that a radius of a circle drawn to the point of tangency is perpendicular to the tangent.]

8. Having obtained the exact answer to Problem 7, can you suggest a good approximate answer which would suffice for many applications and yet make calculation easier?

9. A boy stands on a cliff mi above the sea. How far away is the horizon?

10. Knowing that of all rectangles with the same perimeter, the square has maximum area, prove that of all rectangles with the same area, the square has the least perimeter. [Suggestion: Use the indirect method of proof. Suppose, then, that the square has more perimeter than the rectangle of the same area and consider the square which has the same perimeter as the rectangle.]

Fig. 6–16

6–4 EUCLIDEAN GEOMETRY AND THE STUDY OF LIGHT

Light is certainly a pervasive phenomenon. Man and the physical world are subject daily to the light of the sun, and the process of vision of course is dependent upon light. Hence it is to be expected that the Greeks, the first great students of nature, would investigate this phenomenon. Plato and Aristotle had much to say on the nature of light, and the Greek mathematicians also tackled the subject. It has continued to be a primary concern of mathematicians and physicists right down to the present day. Despite man’s continuous experience with light, the nature of this occurrence is still largely a mystery. Through mathematics and through Euclidean geometry in particular, man obtained his first grip on the subject. Two books by Euclid were the beginning of the mathematical attack.

In ordinary air, light is observed to travel along straight lines. This preference of light for the simplest and shortest path is in itself of significance. But Euclid proceeded beyond this point to study the behavior of light under reflection in a mirror, and discovered a now famous mathematical law of light.

Suppose light issuing from A (Fig. 6–17) takes the path AP to the point P on the mirror m. As we all know, the light is reflected and takes a new direction, PA′. The significant fact about this reflection, which was pointed out by Euclid, is that the reflected ray, i.e., the line PA′ along which the reflected light travels, always takes a direction such that angle 1 equals angle 2. Angle 1 is called the angle of incidence and angle 2 the angle of reflection.* It is, of course, very obliging of light to follow such a simple mathematical law. As a consequence, we are able to prove other facts rather readily.

Fig. 6–17.
The law of reflection of light.

Assume there is a source of light at A (Fig. 6–18), and rays of light spread out in all directions from A. Many of these will strike the mirror. But through a definite point A′ only one of these rays will pass, namely the ray PA′ for which angle 1 equals angle 2. To prove that only one ray from A will pass through A′, let us suppose that another ray, AQ, is also reflected to A′. Now angle 2 is an exterior angle of triangle A′QP. Hence

∠2 > ∠4.

Angle 3 is an exterior angle of triangle AQP, and so

∠3 > ∠1.

Since angle 1 equals angle 2, we see from the two preceding inequalities that

∠3 > ∠4.

Then QA′ cannot be the reflected ray corresponding to the incident ray AQ because the reflected ray must make an angle with the mirror which equals angle 3.

Fig. 6–18

The more interesting point, which was first observed and proved by the Greek mathematician and engineer Heron (first century A.D.), is that the unique ray from A (Fig. 6–19) which does reach A′ after reflection in the mirror travels the shortest possible path in going from A to the mirror and then to A′. In other words, AP + PA′ is less than AQ + QA′, where Q is any point on the mirror other than P, the point at which the angle of incidence equals the angle of reflection.

Fig. 6–19.
The shortest path from A to A′ is the one for which ∠ 1 = ∠2.

How can we prove this theorem? Nature not only sets problems for us, but often solves them too, if we are but keen enough in our observations. If a person at A′ sees in the mirror the reflection of an object at A, he must be looking in the direction A′P and actually sees the image of A at B. Hence, perhaps we should bring B into our thinking. Closer observation shows that the mirror image of an object is on the perpendicular from A to the mirror and, moreover, seems to be as far behind the mirror as the object is in front. That is, AB seems to be perpendicular to the mirror and AC seems to equal CB.

Let us use this suggestion. We construct the perpendicular from A to the mirror, thus obtaining AC, and extend AC by its own length to B. Now it is not hard to see that triangles ACQ and BCQ are congruent because QC is common to both triangles, the angles at C are right angles, and AC = CB. Hence, AQ = BQ, because they are corresponding parts of congruent triangles. Likewise triangles ACP and BCP are congruent and AP = BP. We wish to prove that

(AP + PA′) < (AQ + QA′).

But now, since AP = BP and AQ = BQ, it will be enough to prove that

Well, we have exchanged one difficulty for another, but perhaps this second one is easier to overcome. Physically one looks directly along A′P and sees B. If we could prove that BPA′ is a straight line, then, of course, the inequality (3) would be proved because BQ and QA′ are the other two sides of triangle A′BQ, and the sum of these two sides must be greater than the third side. Our goal, then, is to prove that A′PB is a straight line.

We know that

because m is a straight line. But angle 1 equals angle 4 because triangle PCA and PCB are congruent. Also, according to the law of reflection, angle 2 equals angle 1. If, therefore, in (4) we replace angle 1 by angle 4 and angle 2 by angle 1, we have

Hence, A′PB is a straight line and the inequality (3) is proved. Then the light ray, in going from A to m to A′, really travels the shortest path.

This behavior of light rays is striking. It seems to show that nature is interested in accomplishing its ends by the most efficient means. We shall find this theme to be a recurring one, and it will be seen to have broad applicability.

We have proved a theorem about light rays, but we have also proved somewhat more. As far as the mathematics is concerned, the lines AP and PA′ are any lines which make equal angles with m, and the fact that they are light rays plays no role. What we have proved, then, is a theorem of geometry, namely:

Of all the broken line paths from a point A to a point on a line and then to a point A′ on the same side as A, the shortest path is the one fixed by the point P on m for which AP and A′P make equal angles with m.

This theorem has applications in quite different domains (see the exercises). It is worth noting how the study of light gives rise to purely mathematical theorems. The converse of this theorem is, incidentally, equally true and is presented in the exercises.

EXERCISES

1. Where is the mirror image of a point A which is in front of a plane mirror?

2. Suppose that m (Fig. 6–20) is the shore of a river and a pier is to be built somewhere along m so that merchandise can be trucked from the pier to two inland towns, A and A′. Where should the pier be built so that the total trucking distance from the pier to A and from the pier to A′ is a minimum?

Fig. 6–20

Fig. 6–21

3. A billiard player wishes to hit the ball at A (Fig. 6–21) in such a way that it will strike side m of the table and then hit the ball at A′. Now billiard balls behave like light rays, that is, the angle of reflection equals the angle of incidence. At what point on m should the billiard player aim?

4. A billiard ball starting from a point A on the table (Fig. 6–21) strikes two successive sides and then travels along the table. What can you say about the final in relation to the original direction of travel?

Fig. 6–22

5. In the text we proved that if angle 1 equals angle 2 (Fig. 6–22), then AP + PA′ is the shortest path from A to any point on the mirror to A′. Prove the converse, namely, that if AP + PA′ is the shortest path, then ∠1 must equal ∠2. [Suggestion: Use the indirect method of proof. If ∠1 does not equal ∠2, then one can find another point, P′, on m for which the angles made by AP′ and A′P′ with m are equal.]

6–5 CONIC SECTIONS

The Elements of Euclid dealt with plane figures which can be built up with line segments and circles, with the corresponding solid figures which can be built up with pieces of a plane, such as prisms and the regular polyhedra, and with the sphere. But the classical Greeks also studied another class of curves which they called conic sections because they were originally obtained by slicing a cone with a plane. The resulting curves, the parabola, ellipse, and hyperbola, were treated by Euclid in a separate book. Unfortunately, no copies of this book have survived. But a little after Euclid’s time another famous Greek geometer, Apollonius, wrote a book entitled Conic Sections, which is known to us and which is about as exhaustive in its treatment of these curves as the Elements are about figures formed by lines and circles.

Fig. 6–23.
The parabola.

Fig. 6–24.
The ellipse.

Conic sections were introduced, as already noted, by cutting a conical surface with a plane. However, the curves themselves can be considered apart from the surface on which they lie. For example, the circle is also one of the conic sections. Yet we know that the circle can be defined as the set of all points which are at a fixed distance from a given point, and this definition does not involve the cone at all. Indeed, insofar as properties and applications of these curves are concerned, it is far more convenient to disregard the conical surface and concentrate on the curves themselves.

Let us consider, therefore, the direct definitions of conic sections. To define the parabola, we start with a fixed point F and a fixed line d (Fig. 6–23). We then consider the set of all points, each of which is equally distant from F and d. Thus the point P in Fig. 6–23 is such that PF = PD. The collection of all points, each of which is equidistant from F and d, fills out a curve called the parabola. The point F is called the focus of the parabola, and the line d is called the directrix.

Each choice of a point F and line d determines a parabola. Hence there are infinitely many different parabolas. The general shape of all such curves is, however, about the same. Each is symmetric about the line which passes through F and is perpendicular to d. This line is called the axis of the parabola. Each parabola passes between its focus and directrix and opens out as it extends farther and farther from the directrix.

The direct definition of the ellipse is also simple. We start with two fixed points F and F′ (Fig. 6–24) and consider any constant quantity greater than the distance F to F′. If, for example, the distance from F to F′ is 6, we may choose 10 as the constant quantity. One then determines all points for each of which the distance from F and the distance from F′ add up to 10. This collection of points is called an ellipse. Thus, if P is a point for which PF + PF′ equals 10, then P lies on the ellipse determined by F, F′, and the quantity 10. The points F and F′ are called the foci of the ellipse.

By changing the distance FF′ or the quantity 10, one obtains another ellipse. Some ellipses are long and narrow; others are almost circular. All are symmetric about the line FF′ and about the line perpendicular to and midway between F and F′.

The direct definition of the hyperbola also calls for choosing two fixed points F and F′, called foci, and a constant quantity which, however, must be less than the distance from F to F′. If FF′ is 6, then the constant quantity can, for example, be 4. We now consider any point P for which the difference PF′ – PF equals 4. All such points lie on the right-hand portion of Fig. 6–25, whereas the points for which PF – PF′ = 4 lie on the left-hand portion of the figure. The two portions together are the hyperbola; each portion is a branch of the hyperbola.

Fig. 6–25.
The hyperbola.

As for the ellipse, each choice of the distance FF′ and the constant quantity determines a hyperbola. Here, too, the curve is symmetric about the line FF′ and about a line perpendicular to and midway between F and F′. One branch opens to the right and the other to the left.

We shall not prove that the curves we have defined by means of focus and directrix or by means of foci and constant quantities are the same as those obtainable by slicing a conical surface. In our future work we shall use the direct definitions.

EXERCISES

1. Since the circle is also a conic section, it should be included among one of the three types—parabola, ellipse, and hyperbola. From the shapes of these curves it would appear that the circle falls among the ellipses. Can you see how the circle may arise as a special kind of ellipse?

2. Suppose that we have an ellipse for which F′F is 6 and the constant quantity is 10. If the point P of the ellipse lies on the line F′F to the right of F, how much is PF?

3. For the ellipse, why must the constant quantity be chosen greater than the distance F′F?

4. Given a parabola for which the distance from focus to directrix is 10, how far from the focus is that point on the parabola which lies on the axis?

6–6 CONIC SECTIONS AND LIGHT

Next to straight line and circle, conic sections are the most valuable curves mathematics has to offer for the study of the physical world. We shall examine here the uses of the parabola in the control of light.

Fig. 6–26.
The reflecting property of the parabola.

Fig. 6–27

Let P be any point on the parabola (Fig. 6–26). By the tangent to the parabola at P we mean the line through P which meets the parabola in just that one point and lies entirely outside the curve. From the standpoint of the control of light, the curve possesses a most pertinent property. If P is any point on the curve and F is the focus, then the line FP and PV, the line through P parallel to the axis, where V is any point on this parallel, make equal angles with the tangent t at P. That is, angle 1 equals angle 2.

Before proving the geometrical property just stated, let us see why it is significant. If a light ray issues from some source of light at F and strikes a parabolic mirror at P, it will be reflected in accordance with the law that the angle of incidence equals the angle of reflection. The curve acts at P as though it had the direction of the tangent. Then angle 1 is the angle of incidence. Because angle 1 equals angle 2, the reflected ray will be PV. Hence the reflected ray will travel out parallel to the axis of the parabola. Now, P is any point on the parabola. Hence any ray leaving F and striking the parabola will, after reflection, travel out parallel to the axis of the parabola, and the reflected light will form a powerful beam in one direction. We thus obtain a concentration of light.

Let us now prove that PF and PV make equal angles with the tangent at P. We shall prove first that every point outside of the parabola is farther from the focus than from the directrix, and every point inside is closer to the focus than to the directrix. Consider the point Q (Fig. 6–27) outside the parabola. We wish to show QF > QD, where QD is the distance from Q to the directrix. We continue the line QD until it strikes the parabola at P. Now

QF > PF – PQ

because any side of a triangle is greater than the difference of the other two sides. Since P is on the parabola, by the very definition of the curve, PF = PD. then

QF > PD – PQ = QD.

We may use the same figure to show that Q′, any point inside the parabola, is closer to F than to the directrix, that is, that Q′F < Q′D. First,

Q′F < PF + PQ′

because any side of a triangle is less than the sum of the other two sides. Since PF = PD,

Q′F < PD + PQ′ = Q′D.

And now let us prove that PF and PV of Fig. 6–26 make equal angles with the tangent t at P. We shall invert our approach just to make the proof easier. Let us draw a line t through the point P (Fig. 6–28), which makes equal angles with PF and PV, and we shall prove that this line is the tangent to the parabola at P. We shall make the proof by showing that any point Q on this line lies outside the parabola. Since this line does have one point P in common with the parabola, it must, by the definition of tangent, be the tangent.

Fig. 6–28

Consider the triangles PDQ and PFQ. We know that PD = PF because P is a point on the parabola. Further, since ∠1 = ∠2 by the very choice of the line t, and since ∠2 = ∠3 because they are vertical angles, then ∠1 = ∠3. Finally, PQ is common to the two triangles. Then the triangles are congruent and QD = QF because they are corresponding sides of the congruent triangles. Now QE is the distance from Q to the directrix, and QE < QD because the hypotenuse of a right triangle is longer than either arm. Then QF > QE. According to the preceding proof, Q must lie outside the parabola. Since Q is any point on t (except P, of course), the line t must be the tangent at P. Thus the line through P which makes equal angles with PF and PV is the tangent at P.

We now know, then, that any light ray issuing from F and striking the parabola at P will be reflected along PV, that is, parallel to the axis. The parabolic mirror’s power to concentrate light in one direction is very useful. The commonest application is found in automobile headlights. In each headlight there is a small bulb. Surrounding this bulb is a surface (Fig. 6–29), called a paraboloid, which is formed by rotating a parabola about its axis. (The surface is, of course, silvered so that it will reflect.) Light issuing in millions of directions from the bulb, which is placed at the focus of the paraboloidal mirror, strikes the mirror, is reflected along the axis of the paraboloid, and illuminates strongly whatever lies in that direction. The effectiveness of this arrangement may be judged from the fact that the light thrown forward by bulb and mirror is about 6000 times as intense as that thrown in the same direction by the bulb alone. The reflecting property of the paraboloidal mirror is also utilized in searchlights and flashlights.

The reflecting property of a paraboloidal mirror can be used in reverse. If a beam of parallel light rays enters such a mirror while traveling parallel to the axis, each ray will be reflected by some point on the surface in accordance with the law of reflection. But since FP (Fig. 6–29) and VP make equal angles with the tangent, the reflected ray will travel along PF and all reflected rays will arrive at the focus F. Hence there will be a great concentration of light at F.

Fig. 6–29.
The paraboloidal mirror.

This concentration of light is used effectively in telescopes. The light emitted by stars is so faint that it is necessary to collect as much as possible in order to obtain a clear image. The axis of the telescope is therefore directed toward the star, and, because this source is so far away, the rays enter the telescope practically parallel to the axis, travel down the telescope to a paraboloidal mirror at the back, and are reflected to the focus of the mirror.

Radio waves behave very much like light rays. Hence paraboloidal reflectors made of metal are used to concentrate radio waves issuing from a small source into a powerful beam. Conversely, a paraboloidal antenna can pick up faint radio signals and produce a relatively strong signal at the focus. Since radio is used today for hundreds of purposes, the paraboloidal radio antenna is a very common instrument.

We see from this brief account that the conic sections are immensely valuable. Some of the most momentous applications have yet to be described and will be taken up in later chapters.

How did the Greeks come to study these curves? As far as we know, the conic sections were discovered in attempts to solve the famous construction problems of Euclidean geometry, i.e., to trisect any angle, to construct a square equal in area to a given circle, and to construct the side of a cube whose volume is twice that of a cube of given side. The constructions were to be performed subject to the restriction that only a straight edge (not a ruler) and a compass be used. Having obtained the curves, the Greeks continued to work on them, partly because they were interested in geometrical forms and partly because they discovered the uses of these curves in the control of light. Apollonius himself wrote a book entitled On Burning Glasses, whose subject was the parabola as a means of concentrating light and heat, and there is a story that Archimedes constructed a huge paraboloid which focused the sun’s rays on the Roman ships besieging his city of Syracuse, and thus set them on fire.

We see in the history of conic sections one more example of how mathematicians, pursuing a subject far beyond the immediate problems which give rise to it, come to make important contributions to science.

EXERCISES

1. Let Q be any point outside of an ellipse (Fig. 6–30). Prove that F₂Q + F₁Q is greater than a, where a is the sum of the distances of any point on the ellipse from the foci. [Suggestion: Introduce the point P where F₂Q cuts the ellipse.]

Fig. 6–30

Fig. 6–31

2. Let t be the tangent at any point P of an ellipse (Fig. 6–31). Let F₂ and F₁ be the foci. Prove that F₂P and F₁P make equal angles with t. [Suggestion: Use the result of Exercise 1 and Exercise 5 of Section 6–4.]

3. In view of the result of Exercise 2, what do you expect to happen to the light rays issuing from a source placed at the focus F₂ of the ellipse?

4. When the distance between the two foci F₂ and F₁ of an ellipse approaches 0, the ellipse approaches a circle in shape. What do the lengths F₂P and F₁P become when F₂ and F₁ coincide? What theorem about circles follows as a special case of the result in Exercise 2?

6–7 THE CULTURAL INFLUENCE OF EUCLIDEAN GEOMETRY

If the development of mathematics had ceased with the creation of Euclidean geometry, the contribution of the subject to the molding of Western civilization would still have been enormous, for Euclidean geometry was and still is an overwhelming demonstration of the power and effectiveness of our reasoning faculty. The Greeks loved to reason and applied it to philosophy, political theory, and literary criticism. But philosophy breaks down into philosophies whose relative merits become the object of much dispute between the adherents of one school and those of another. Plato’s Republic may indeed be the perfect answer to the quest for a satisfactory political system, but we must still be convinced of this fact. And literary criticism certainly does not lead to universally accepted standards and the creation of universally acclaimed literature. In Euclidean geometry, however, the Greeks showed how reasoning which is based on just ten facts, the axioms, could produce thousands of new conclusions, mostly unforeseen, and each as indubitably true of the physical world as the original axioms. New, unquestionable, thoroughly reliable, and usable knowledge was obtained, knowledge which obviated the need for experience or which could not be obtained in any other way.

The Greeks, therefore, demonstrated the power of a faculty which had not been put to use in other civilizations, much as if they had suddenly shown the world the existence of a sixth sense which no one had previously recognized. Clearly, then, the way to build sound systems of thought in any field was to start with truths, apply deductive reasoning carefully and exclusively to these basic truths, and thus obtain an unquestionable body of conclusions and new knowledge.

The Greeks themselves recognized this broader significance of Euclidean geometry, and Aristotle stressed that the Euclidean procedure must be the aim and goal of all sciences. Each science must start with fundamental principles relevant to its field and proceed by deductive demonstrations of new truths. This ideal was taken over by theologians, philosophers, political theorists, and the physical scientists. We shall see later on how widely and how deeply it influenced subsequent thought.

By teaching mankind the principles of correct reasoning, Euclidean geometry has influenced thought even in fields where extensive deductive systems could not be or have not thus far been erected. Stated otherwise, Euclidean geometry is the father of the science of logic. We pointed out in Chapter 3 that certain ways of combining statements lead to unquestionable conclusions, provided the original premises are unquestionable. These ways are called principles or methods of deductive reasoning. Where did we get these principles? The answer is that the Greeks learned to recognize them in their work on Euclidean geometry and then appreciated that these principles apply to all concepts and relationships. If one argues from the premises that all bankers are wealthy and some bankers are intelligent to the conclusion that some intelligent men are wealthy, he is using a principle of valid reasoning discovered in the work on Euclidean geometry. The indirect method of proof which we applied earlier in this chapter owes its recognition to the same source. Toward the end of the classical Greek period, Aristotle formulated the valid principles of reasoning and created the science of logic. In particular, he called attention to some basic laws of logic, such as the principle of contradiction, which says that no proposition can be both true and false, and the principle of the excluded middle, which states that any proposition must be either true or false.

It is because Euclidean geometry applies these principles of reasoning so clearly and so repeatedly that this subject is often taught as an approach to reasoning. The Greeks themselves stressed the value of mathematics as a preparation for the study of philosophy. Whether this is the best way of learning to reason may perhaps be disputable, but there is no doubt that historically this is the way in which Western man learned. And it is pertinent that even current texts on logic use mathematical examples quite freely because these illustrate the principles clearly, unobscured by irrelevant implications or by vagueness in the concepts and relations employed.

The most portentous fact about Euclidean geometry is that it inspired a large-scale mathematical investigation of nature. From the outset the geometrical studies were an investigation of nature. But as the Greeks proved more and deeper theorems and these theorems continued to agree perfectly with observations and measurements, the Greeks became convinced that through mathematics they were learning some of the secrets of the design of this world. It became clear that mathematics was the instrument for this investigation, and the results fostered the expectation that the further application of mathematics would reveal more and more of that design. Just how far the Greeks were emboldened to carry this venture will be apparent in the next two chapters. From the Greeks the Western world learned that mathematics was the extraordinarily powerful instrument with which to explore nature.

REVIEW EXERCISES

1. One of the basic theorems of Euclidean geometry is that the base angles of an isoceles triangle are equal. Euclid’s proof proceeded thus. Given triangle ABC (Fig. 6–32) with AB = AC, prolong AB to D and AC to E so that BD = CE.

Fig. 6-32

Fig. 6-33

Now draw DE, BE, and DC. Complete the proof by first proving that BE = DC and then that ∠ CBD = ∠ BCE.

2. In the text we proved that if the alternate interior angles 1 and 2 of Fig. 6–33 are equal, the lines AB and CD are parallel. Prove

a) if the corresponding angles 2 and 3 are equal, the lines are parallel;

b) if the angles 2 and 4 are supplementary, that is, if their sum is 180°, then the lines are parallel.

3. Suppose that m in Fig. 6–34 represents a road, and a telephone central is to be built somewhere on this road to serve towns at A and A′. Where along the road should the central be built to minimize the total distance from the central to A and the central to A′?

Fig. 6-34

Fig. 6-35

4. A ship must pass between the guns of a fort at A (Fig. 6–35) and equally powerful guns along the shore m. What path should it take to be as safe as possible from all the guns?

Fig. 6-36

5. Let C and D be two fixed circles (Fig. 6–36) with radii c and d, respectively, and c > d. Moreover D and C are tangent internally at P. Now let T be a third circle which is tangent externally to D and internally to C. Show that the positions of the centers of all possible circles T is an ellipse whose foci are the centers of C and D.

Topics for Further Investigation

1. The use of Euclidean geometry in the design of spherical mirrors. Use the first reference to Kline or the reference to Taylor below or any college physics text.

2. The use of Euclidean geometry in the design of optical lenses. Use the references to Taylor below or any college physics text.

3. The contents of Euclid’s Elements. Use the reference to Heath.

4. Euclidean geometry as a manifestation of Greek culture. Use the second reference to Kline.

THE NATURE AND USES OF EUCLIDEAN GEOMETRY

EXERCISES

EXERCISES

EXERCISES

EXERCISES

REVIEW EXERCISES

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