California State University, Northridge

CALIFORNIA STATE UNIVERSITY, NORTHRIDGE

THE HISTORY OF MATHEMATICS AS A TEACHING TOOL

A graduate project submitted in partial satisfaction of the requirements for the degree of Master of Arts in Secondary Education by Nancy Helen Bayer

June, 1981 Cha~les H. Heimler, Ed.D.

Hughes-,~P~h~.=o-.,~C~ha-.i~r-ma_n

California State University, Northridge

·i i To my loving parents. Thank· you.

iii PREFACE

1 -!,ave more than an impression - it a~2unts to a certainty - that algebra is made repellent by the unwillingness or inability of teachers to explain why . . . . There is no sense of his tory behind the teaching, so the feel ing is given that the whole system dropped down ready-made from the skies, to be used only by born jugglers. (Barzun, 1945, p. 82) A knowledge of the history of mathematics can be a significant tool in the hands of a mathematics teacher. One of the best ways to stimulate curiosity and to instill a spark of th~ fascination of mathematics is to relate some of the absorbing stories with which the history of mathematics abounds.

I have always tncorpOl~ated some math history in my lessons, but until three years ago my mathematical tool kit, like that of many teachers, contained very little historical material. It was during the 1977 - 1978 school year, while on sabbatical leave from teaching mathematics at Santa r1onica Hi"gh School, that I took a course from Dr. Barnabas B. Hughes tn the htstory of mathematics. Today my mathematical tool kit is a lot fuller. It contains many historical appreciations - lessons, stories, biographies of famous mathematicians, word derivations, etc, - that can, if properly used, clarify meanings, increase understandings create interest, and motivate both my students and me. r developed this booklet because I needed to organize my tool

iv kit, to formalize my presentation of historical enrichment material. Each chapter of my project should be specific and complete enough to make historical topics readily available for classroom use. All of the topics are appropriate for a first-year geometry course. Many are equally well suited to other courses in mathematics taught at the high school level. Unfortunately many teachers of mathematics have very little background in the history of mathematics. At most they may have I only a passing acquaintance with some familiar and often inaccurate anecdotes. about famous mathematicians. Compounding the dilemma is a lack of available resources on mathematical history designed for immediate classroom use. Recognizing this need, the National Council of Teachers of Mathematics commissioned a project in 1963 which resulted ih the publication of the thirty-first yearbook,

Histor!~~ IQQic~ For I~ Mathematics Classroom (1969). It is an excellent resource, with concise articles covering many topics in mathematics. Another book I found valuable in my research is A Sut'Vei: Of Nathematics: f.l_eme_iltary Concepts And Their Historical

Dev.~_l9pment (1968) by Vivian ShaioJ Groza. This book is the result of amplifications and developments of lecture notes for a general education course in mathematics. This course is designed as a terminal math class for students with a liberal arts ~ackground.

The writer's objective is to develop an intet~est in mathematics in students who previously had little or none. Both Groza's book and the NCTM publication present many topics in a form that is readily adaptable to classroom use.

\' There are also those works that are classics on mathematical history, books that cover almost any topic and/or mathematician. If a teacher wished to equip oneself with a comprehensive resource, I would suggest An Introduction To The History Of Mathematics (1976) by Howard Eves. In addition to the historical narrative, Eves provides an extensive list of problem studies at the conclusion of each chapter. The problem studies are designed to educate and to motivate high school and college students, as well as their teachers. Another valuable reference is David Smith's History Of Mathematics (2 volumes). Volume I is a chronological survey, while Volume II is arranged topically. In my research, I found only one work similar to mine, a masters project written by Edward Crosson at California State University, Northridge in 1973. The project, Teacher's Guide For Algebra I: A Source Book In Mathematical History, is a collection of historical sketches and problem studies. It is interesting to note that Mr. Crosson and I were inspired by the same man, Dr. Barnabas Hughes. I share Glaiser's conviction that "no study loses more than mathematics by any attempt to dissociate it from its history." I know that my effort; The H·i~tory Of f~athematics As ~ Tea~ing Tool will be a valuable resow·ce for both my students and me. I hope that others will find it useful in their classrooms as well .. I would like to expl·ess my appreciation to Dr. Barnabas B. Hughes, \'Jho has been both my teacher, my advisor, and my inspiration. His guidance, encouragement, and expertise have been invaluable,

vi TABLE OF CONTENTS

Page PREFACE iv ABSTRACT ix

INTRODUCTION 1

CHAPTER

1 THE GEOMETRY OF THE ANCIENTS 7 Exercises For Students 11 References 12

2 ANGULAR MEASURE 14 Exercises For Students 16 References 17 3 THALES OF MILETUS 18 Exercises For Students 20 References 22

4 PYTHAGORAS AND THE PYTH,I\GOREANS 23 Exercises For Students 29 References 31

5 PYTHAGOREAN TRIPLES 32 Exercises For Students 36 References 37

vii CHAPTER Page 6 GEOMETRIC CONSTRUCTIONS 38 Exercises For Students 40 References 41

7 THE GEOt~ETRY OF THE FIXED-COMPASS 42 Exercises For Students 46 References 49 .8 THE FIXED-COMPASS AND REGULAR POLYGON CONSTRUCTIONS 50 Exercises For Students 58 References 60 9 ARCHIMEDES 61 Exercises For Students 67 - References 69

BIBLIOGRAPHY 71 ABSTRACT

THE HISTORY OF MATHEMATICS AS A TEACHING TOOL by Nancy Helen Bayer Master of Arts in Secondary Education

This project is designed to supply teachers and students with a usable collection of topics on the history of mathematics, All of the topics selected are appropriate as enrichment material for a first-year geometry course. They can be keyed to any geometry text. Many of the chapters are suitable for use in other high school mathe matics courses as well. The first chapter deals with the origins of geometry in ancient Egypt and Babylonia. History also answers such questions as why there are 360 degrees in a circle, which is the subject of chapter two. The booklet also contains biographies of famous mathematicians. Complete chapters are devoted to the lives and contributions of Thales, Pythagoras, and Archimedes. The Pythagorean society in general and the derivation of formulas that generate Pythagorean

1X triples in particular are also included as topics. Finally, three chapters have been dedicated to geometric constructions. Euclid's "collapsible compass" geometry is followed by two chapters on the geometry of the fixed-compass and its rich history. An extensive list of exercises for students follows each chapter. Topics introduced are developed with sufficient depth so that problems can be solved in authentically historical settings. In solving problems as Thales, Pythagoras, Euclid, or Archimedes would have, students will more thoroughly understand their methods and should achieve a deeper appreciation of their accomplishments.

It is also my hope that students will gain in their understanding of "today's mathematics" through analyzing older and alternative approaches.

X INTRODUCTION

Topics in mathematics which clarify mathematical meanings, which develop mathematical appreciations, and which furnish opportun ities for the student to discover mathematical ideas provide stimu lating material for both teacher and student. If the topic has an interesting or significant historical development as well, it has an even greater contribution to offer. The opportunities for in cluding iome mathematical history in the instructional program are many and varied. All of the topics selected for this project are appropriate as enrichment material for a first-year geometry course. They can be keyed to any geom2try text. Many of the chapters are suitable for use in other high school mathematics courses as well. Although some topics can be covered in a single day, others may take two or even three days.

The name "geometry 11 is derived from two Greek words meaning "earth measure.'' In chapter one, The Geometry Of The Ancients, we learn that mathematics arose from practical necessity. The Egyptians and Babylonians were interested in geometry because it could be used to solve their daily problems. Their concern was with how to obtain results rather than why those results occur. Efforts ~ere applied toward specific problems, such as surveying land or taking building measurements, with no movement toward abstraction and, generalization. History also answers such questions as why there are 360 degrees in a circle, why there are sixty minutes in a degree or an

1 2

hour, and \'lhy sixty seconds in a minute. In chapter t\'IO~ the Babylonians are credited with our system of angular measure. Stories about the lives of famous mathematicians can be used to create interest and motivate students. According to tradition, demonstrative geometry began with Thales of Miletus in the first half of the sixth century B.C. Thales was the first Greek to lay down guidelines for the development of geometry in abstract terms. Chapter three is the story of Thales' life and his contribution to mathematics. The next outstanding Greek geometer is Pythagoras, the subject of chapter four. The mathematics of Pythagoras and the Pythagorean society has connections with many of the topics we teach in high school mathematics courses. For example, when a student hears how long tt took for· the concept of irrational numbers to be accepted, used, and understood, he does not feel quite so concerned that the concept did not come to him easily. Students are fascinated by the legend surrounding the discovery of irrational numbers by the

Pythagoreans, The Pythagoreans believ~d that the entire universe could be expressed in terms of ratios of whole numbers or in tel~ms of whole numbers themselves. Understandably, they were shocked, like students are today,_ wh.en they evaluated the diagonal of a unit square, But for the Pyt~agoreans it was as much of a .shock as the revelation that there was no God would be to a Christian. The best remembered of the Pythagorean teachings is the Pythagorean theorem. Closely allied to the theorem is the problem of finding Pythagorean triples. The history of the problem of 3

finding formulas that generate Pythagorean triples is the subject of chapter five. Students are amazed to learn that evidence seems good that the Babylonians had a general and systematic solution for the problem of Pythagorean triples. Plimpton 322, a cuneiform tablet discovered in 1945, lists a series of fifteen sets of Pytha gorean triples. Plimpton 322 belongs to the period 1900 B.C. to 1600 B.C. and is the first recorded example of work in number theory; The lesson on Pythagorean triples takes a heuristic approach as well as a historic one. With a little encouragement and guidance, students .can get enormous satisfaction out of discovering and veri fying the formulas for Pythagorean triples on their own. It seems to me that geometric constructions can not be taught without telling students that· Plato is credited with restricting the geometer to the ~se of straightedge and comp~ss alone, and that

Euclid 1 s geometry was based on constructions carried out by the straightedge and .. collapsible compass ... But it has been my observa tion that this topic is handled very poorly in most geometry texts and by many geometry teachers. Geometric constructions have such great potential for motivating students, yet students complain about what seem to be the arbitrary steps they must follow. In chapter six, there is a sense _of history behind the teaching. Another point of interest for the study of geometrical con struct ions is that of 11 construct ions vJith 1 imited means. 11 Students are intrigued when they learn that there are geometries which pre suppose the use of tools other than the rigid compass they use. Fixed-compass or rusty-compass geometry makes a fascinating 4 discovery lesson, and it's history is of particular interest and value. It is associated with Abu'l-Wefa, an Arab mathematician of the tenth century. It was also a tool of artists of the fifteenth and sixteenth centuries, notably Albrecht Durer and Leonardo da Vinci. Chapters seven and eight are devoted to the geometry of the fixed compass. My final chapter is the story of Archimedes, regarded,as the greatest mathematician and scientist of the ancient world. Students will be intrigued by his many scientific discoveries and mechanical inventions. Exercises for students are listed at the conciusion of each chapter. The lists are extensive and the exercises are of varying degrees of difficulty. This will enable the teacher to se1ect problems that fit students' interests and abilities. I recommend discussing a number of the problems in class, and assigning others to be worked out at home. The purpose of the exercises is to reinforce and to motivate. It is also my hope that students will gain in their understanding of "today's mathematics~ through analyzing older and alternative approaches. Many of the exercises concern themselves with histori cally important problems and procedures; others are purely recre ational. Some of the exercises can lead to short research papers, particularly those included in the chapter on Pythagorean mathematic~ Topics introduced are developed with sufficient depth so that the student can solve problems in an authentically historical setting. Often he is asked to make comparisons with modern methods. 5

For example, one exercise has the student compute the area of a circle using our modern formula as well as the inexact Egyptian and Babylonian formulas. Another exercise challenges him to devise a technique for determining the height of a pyramid from the length of its shadow as Thales would have done. In the unit on Pythagorean triples, the student is asked to produce triples using the formulas attributed to Pythagoras, Plato, and Euclid. The construction exercises in chapter six are carried out with Euclid's collapsible I compass. In chapters seven and eight the compass is instead used as if it were a fixed-compass. Many of the fixed-compass problems are the same regular ·polygon constructions used by Albrecht Durer and Leonardo da Vinci. The interested student or teacher will want to consult further literature. Accordingly, each chapter is followed by a list of references dealing with the material of that chapter. Additional information, examples, and problems may be found throughout the literature cited, An additional general bibliography, given after the final chapter, applies to almost every chapter. The history of mathematics can be a useful and effective teaching tool. It may be that a brief historical talk or anecdote is indicated. Or historical material may be used to introduce a topic or to encourage student discovery. Other historical stories and topics can provide enrichment material fdr reports, projects, independent study, and club progr-ams. "Teaching so that students understand the whys, teaching for mean1ng and understanding, teaching so that children see and appreciate the nature, role and 6 fascination of mathematics'' is our challenge. (NCTM, 1969, p. 1) 6

fascination of mathematics .. is our challenge. (NCTM, 1969, p. 1) Chapter 1

THE GEOMETRY OF THE ANCIENTS

The Greek historian Herodotus writes . They said also that this King {Sesostris) divided the land among all Egyptians so as to give each one a quadrangle of equal size and to draw from each his revenues, by imposing a tax to be levied yearly. But every one from whose part the river tore away anything, had to go to him and notify what had happened; he then sent the overseers, who had to measure out by how much the land had become smaller, in order that the owner might pay on what was left; in proportion to the entire tax imposed. ·In this way, it appears to me, geometry originated, which passed thence to Hellas. (Cajori, 1919, p. 9) Geometry is said to have originated in the surveying done by the Egyptians over 4000 years ago. The annual inundation of the Nile Valley forced the Egyptians to develop some system of redetermining land markings. In fact, the name 11 geometry" is derived from two Gt·eek words - "geo" meaning "earth," and ''metron" meaning "measure." A need for the development of surveying techniques was especially imperative if, as Herodotus stated, taxes in Egypt were paid on the basis of land area. The Egyptian surveyors did not h~ve accurate measuring instruments, however, and faulty measw~ements often 1 ed to incorrect rules and formulas for determining the size of a piece of land. The Babylonians likewise encountered an urgent need for geometry. Marsh drainage, irrigation, and flood control made it

7 8

possible to convert the land along the Tigris and Euphrates rivers into a rich agricultural region. The Babylonians were highly skilled irrigation engineers. They were also remarkable builders. The temples and palaces of ancient Babylon, and the Egyptian pyramids could not have been built without the knowledge of and use of geometry. The geometry of Egypt and Babylon was of the rule-of-thumb or practical variety. Their formulas for areas of land and volumes of ' granaries were arrived at by trial and error. Both Egyptians and Babylonians had the correct formulas for the areas of the square, rectangle, certain simple triangles, and right trapezoids. But many other formulas were incorrect. The Babylonians used the . s . . incorrect formula A = -(a + b) instead of the correct formula 2 A = 2h (a + b) for calculating the area of the isosceles trapezoid.

The same incorrect formula for the area of a quadrilateral was used

·by both the Egyptians and Babylonians:

K = (a + c)(b + d) where 4

a, b, c, and dare consecutive sides. This formula gives the correct result only if the quadrilateral is a rectangle. In every other case the formula gives too large an answer. g

The Egyptians also used an incorrect formula for computing the 2 64d area of a circle: = ---sr- This is equivalent to using

the approximation 7f = 3.1604 ..• An even less accurate value of n is implied by the Babylonian approximations for the 1 2 circumference and area of a circle: . C = 3d and A = -12 c This is equivalent to using the approximation 3 for n. It is ipteresting to note that this value for n is also found in the Bible (I Kings 7:23). Both the Egyptians and the Babylonians found the volume of a right circular cylinder by multiplying the area of the circular base by the height. This procedure is correct, but their answers were in error due to their incorrect circle formulas.

2 The Babylonians used v = t d h i h 2 t The Egyptians used .v = ~i d h lrd 41 2 2 and we use = 7f d h (or nr h) with 7f = 3. 141 64. v 4

The Egyptians and Babylonians also knew the correct formulas for calculating volumes of the cube and the box. Cubes, boxes, and cylinders were used as containers for grain. The Egyptians even knew the correct formula for the volume of a 2 truncated pyramid. Its volume is given as -}h (a2 + b + ab), in Problem 14 of the Moscow papyrus. This problem illustrates the highest point of Egyptian geometry. Eric Temple Bell, a renowned historian of mathematics, has called the solution 11 the greatest 10

Egyptian pyramid. 11 i h 1

The geometry of ancient Egypt and Babylonia was motivated by practical needs and developed by empirical processes. While some rather remarkable geometrical results were obtained, these results were judged mainly by their usefulness in practical situations. Efforts were applied to specific problems, such as surveying land or taking building ~easurements. No movement was made toward abstraction or generalization. This concept of utilitarianism gradually changed with economic and political changes. The Egyptians and Babylonians lost their leadership in scholarly pursuits and the Greeks eventually assumed the role. About the seventh century B.C. an active commer·cial intercourse sprang up between ~gypt, Babylonia and Greece. Naturally there arose an interchange of ideas as well as merchandise. While geometry started out being much like surveying, the Greeks developed it into something very different. The Greeks were the real inventors of what we now call geometry. The Egyptians and Babylonians carried geometry no further than was absolutely necessary for their practical wants. The Greeks, on the other hand, had 11

within them a strong speculative tendency. They wanted to know why certain results were obtained in addition to knowing how they were obtained. With the Greeks, a new and refreshing emphasis was placed on theory. Their logical, deductive approach was a major break- through. With it the Greeks discovered things that the crude experi mental methods of the Egyptians and Babylonians would never have revealed.

EXERCISES FOR STUDENTS 1. The Babylonians used the incorrect formula A= I(a+b) instead

h of the correct formula A = - (a+b) for calculating the area of an 2 isosceles trapezoid. Using the figure below, calculate the area of the trapezoid by the incorrect Babylonian method and by our modern formula. 5

I 119.2 I

16.2 2. Show that the ancient Egyptian and Babylonian procedure for finding the area of a quadrilateral gives a correct result if the figure is a rectangle and gives too large a number if the figure is a nonrectangular parallelogram or a trapezoid. Is this procedure ever correct for a quadrilateral that is not a rectangle? 3. Find the difference in the number of feet in the circumference, C, of the base of a circular storage tank as computed by an inexact

Babylonian formula, C =3d, and the exact formula, C = 1rd, if 12

the diameter is 10 feet. (Use 3.14 as an approximate value for TI • ) 4. Find the difference in the number of square feet in the area, A, of the base of a circular storage tank as computed by an inexact 2 2 64d A = nd Egyptian formula, A = ---sl and the exact formula, 4 if the diameter is 10 feet. (Use 3.14 as an approximate value for n.) 5. Use exercise 3 and 4 to compare ancient Babylonian and Egyptian values for 1T. 6. Compute the volume of a truncated square pyramid of vertical height 9 units, upper base 3 units, and lower base 5 units by using the correct Egyptian method,

7. Compute the area of a circle with diameter d, 18 units (radius r, 9 units} by using the a. Babylonian method, A= 3r2. · .64d 2 b. Egyptian method, A = -sl

c. Modern method, A= nr2 with n = 3.1416.

REFERENCES Bunt, Lucas, Phillip Jbnes, and Jack Bedient, The Historical Roots Of Elementar,y_ Mathematics (New Jersey: Prentice-H.all, Inc., 197~p. 33-40, 58-63.

Cajori, Flor~ian, ~ Histor.:_y of Mathematics (New York: The ~1acmil1an Company, 1929), pp. 9-15, Eves, Howard and Carroll Newsom, An Introduction To The Foundations And Fundamental Con~ts Of ·Mathematics (New--York: Holt, Rinehart and Winston, l964T, pp. l-11 . 13

Groza, Vivian ShavJ, fl Survay Of ~~athematics (NevJ York: Holt, Rinehart and Winston, 1968), pp. 83-93. Kline, Morris, Mathematics In Western Culture (New York: Oxford University Press, 1953), pp. 13-23. Chapter 2

ANGULAR r1EASURE

Ancient Babylonians deserve credit for our present division of the circumference of a circle into 360 equal parts. Thei\ civiliza tion started about 4000 B.C. when they drained marshes, cultivated fields, built cities, and exchanged goods. Thereafter they developed an interest in astronomy. It had two purposes: one in relation to religion, the other for constructing a calendar. The latter deter mined the yearly cycle of planting and harvestin~. The Babylonians also developed a sexagesimal number system, a system with 60 ~s its base. Whole numbers ·and fractions were written jn a positional notation system. (The idea of a decimal point and positions to its right representing tens, hundreds, etc., did not enter our Hindu-Arabic number system until about 1585, over 4000 years later.) Although 60 is an extraordinarily large number to use as the base of a number system, we still use it every day in our division of an hour into 60 minutes, of a minute into 60 seconds, and of a circle into six times 60 aegrees. Why the Babylonians chose 60 no one knows. Several explanations have been suggested. As any student of geometry knows, a circle can be divided into six equal parts using its radius as a chord. The choice of 60 may have come from l/6 of 360. Another guess is that 60 was chosen because of its numerous integral divism~s- 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30,

14 15

and 60, which would simplify work with fractions. Since the Babylon iftns were accomplished astronomers, the sexagesimal system could have had an astronomical origin. The division of the circumference of a circle into 360 equal parts may have originated from their slightly inaccurate estimation of 360 days in a year. However, since the normal development of a number system is from lower to higher units, it seems more likely that the sexagesimal number system wfth base 60 preceded the division of the circle into 360 equal parts. Through contacts of trade and conquest, the Greek civilization partially absorbed the Babylonian culture. Greek astronomers adopted the sexagesimal number system for writing fractions. Fractions were written using a place value notion much as we write decimal fractions. For example, one-half and one-third were designated by 30 and 20, the reader being expected to supply the v.Jord "sixtieths." Hypsicles (c. 180 B.C.) was the first Greek astronomer to divide the circle of the zodiac into 360 parts, following the Chaldeans who had divided it into 12 signs and each sign into 30 parts. Hipparchus (c. 150 B.C.) is credited with generalizing the idea to all circles. The famous Greek astronomer and geographer, Ptolemy (c. 125 B.C.), used sexagesimal fractions in all t~pes of computation. However, it was not until Theon of Alexandria (c. 350 A.D.) that sexagesimal fractions were used for measuring time. Babylonian sexagesimal frar.tions as absorbed by Greek astronomers were adopted by succeeding Arabic civilization. When Greek astron omers treatises were translated into Arabic, their system for writing fractions and their terminology was retained. Sixtieths were called 16

11 first small parts 11 and sixtieths of sixtieths were called 11 Second small parts. 11 In Latin these phrases became 11 pars minuta priman and

11 pars minuta secunda 11 from which we got our 11 minutes 11 and 11 Seconds. 11 So we know that our division of the circle into 360 degrees, each degree into 60 minutes, and each minute into 60 seconds undoubtedly dates back to the ancient Babylonians, as does our division of the hour into 60 minutes and the minute into 60 seconds. This system of angular measure is the most popular, and as we have seen, the reasons for its development are purely historical.

EXERCISES FOR STUDENTS 1. Create your own division of the circle; for example, by clividing it by 15. 2. What would be the effect of a circle divided by 10, to produce a decimal system?

3. Another interesting accoun~ for our division of the circle into 360 equal parts has been advocated by Otto Neugebauer. In early Sumerian times there existed a large distance unit, a sort of Baby lonian mile, equal to about seven of our miles. Since the Babylonian mile was used for measuring longer distances, it was natural that it should also become a t~me unit, namely the time required to travel a Babylonian mile. When Babylonian astronomy reached the stage in which systematic records of celestial phenomena were kept, the Babylonian time-mile was adopted for measuring spans of time. Since a complete day was found to be equal to 12 time-miles, and one com plete day is equivalent to one revolution of the sky, a complete 17 circuit was divided into 12 equal parts. But, for convenience, the

Babylonian mile had been subdivided into 30 equal parts.

Show that by this account we would arrive at 360 equal parts in a complete circuit.

REFERENCES Bunt, Lucas, Phillip Jones, and Jack Bedient, The Historical Roots Of Elementary Mathematics (New Jersey: Prentice-Hall, Inc., 1976), p. 47.

Cajo~t, Florian, A History Of Mathematics (New York: The Macmillan

Company, 1929) 3 p. 6.- Eves, Howard E., An Introduction To The History Of Mathematics (New York: Holt, Rinehart and Winston, 1976), p. 33, Jones, PhillipS., "Angular Measure- Enough Of Its History To Improve I'ts Teaching," The Mathematics Teacher 1953, 46: 419-426. Smith, David Eugene, History Of Mathematics, Vol. I (New York: Dover Publications, Inc., 1951), p. 41. Chapter 3

THALES OF MILETUS

In the beginning, geometry was mostly elementary and practical. The Egyptians and Babylonians were interested in mathematics because it could be used to solve their daily problems. Their concern was with how to obtain results rather than why those results occur. Efforts were applied to specific problems, such as surveying land or taking building measurements, with no movement toward abstraction and generalization. According to tradition, demonstrative geometry began with Thales of ~1iletus, one _of the "seven wise men" of ·antiquity, during the first half of the sixth century B.C. Called the father of Greek mathemat ics, he spent the early part_of his life as a merchant. He probably visited Egypt, Babylonia, Crete, and Asia. Thales brought the mathe matical knowledge of the ancient Orientals to the Greeks, but more important than this were his own original contributions to the field. Thales was the first Greek to lay down the guidelines for the development of geometry in abstract terms, apart from any practical use to which it might be put. The 11 How?" of the Egyptians and Baby lonians was no longer enough. The Greek man also wanted to know

11\~hy?"

Thales worded properties of geometric figures as general state ments. He is credited v.Jith demonstrating the fol1ovling elementary

18 19 geometrical results: 1. The base angles of an isosceles triang1e are equal. 2. If two straight lines intersect, the vertical angles are equal. 3. Two triangles are congruent if two angles and the included side of one are congruent respectively to two angles and the included side of the other. 4. A circle is bisected by a diameter. 5. An angle inscribed in a semicircle is a right angle. The-theorems Thales pondered were qu·ite simple, but he is said to have been the.first to support them with some logical reasoning or proof. Thales• method of proof was intuitional, rather than proof in the modern sense. For example, he may have proven that semi-circular portions of the same circle are equal by folding a circle along its diameter. Likewise, by folding an isosceles tri angle along the altitude from the vertex angle, he may have demon strated that the base angles are equal. Thales was a merchant in his younger days, a statesman in his middle life, and a mathematician, astronomer, and philosopher in his later years. Four anecdotes describe his wisdom and cleverness. Thales was a shrewd merchant. He once predicted an unusually rich crop of olives, and then quietly bought all available oil presses. Having cornered the market, he made high profits by renting the presses to former owners. Thales was also an ingenious animal trainer. It seems that one of his mules once slipped while carrying sacks of salt across a 20 p •

stream. Since much of the salt dissolved in the stream, the mule•s load was considerably lightened. So the smart mule repeated the slipping. But on the next trip Thales loaded the mule•s sacks with rags and sponges. They absorbed the water and thus increased the mule•s load. This remedy soon cured the smart mule of a bad habit. The study of scientific astronomy also began with Thales. He is said to have predicted a solar eclipse in 585 B.C. Recent research, however, indicates no evidence of this. One day, according to legend, Thales fell into a ditch while observing the stars. An old woman passing by asked how he could hope to know what was happening in the heavens when he couldn•t see what was at his feet. Thales was also an applied mathematician. In Egypt he aston ished King Amasis by determining the height of a pyramid from the length of its shadow. He is also supposed to have used the A.S.A. theorem to find the distance of a ship from shore.

EXERCISE FOR STUDENTS l. 11 Pons asinorum 11 or 11 bridge of asses 11 is a name commonly used for

Proposition 5 of Book I of Euclid 1 s Elements: The base angles of an isosceles triangle are equal to one another. Thales is credited with the discovery of the theorem. Some believe that the term "bridge of asses 11 arose during the Dark Ages. Students in European universities studied Latin translations of Euclid 1 s book in order to get their degrees. Unfortunately, when some of these students reached Thales• theorem, they hesitated or even decided to quit. Students for whom the theorem was an obstacle were called dunces; hence, the theorem 21

11 11 was given the nickname, bridge of asses • Give a modern proof of this theorem. 2. Determine how Thales might have proved by paper-folding, the theorem: Vertical angles are equal.

3. Historical accounts from Hieronymus, a pupil of Aristotle, indicate that Thales measured the heights of pyramids by measuring their shadows at the time when a body and its shadow were equal in length. An account by Plutarch credits Thales with using the more general method of similar triangles, by setting up a stick at the end of the shadow. Devise a method for determining the height of a pyramid based on this more general method of similar triangles. 4. Proclus states that Thales used the A.S.A. theorem to determine the distances of ships at sea. How Thales did this is not known for certain, but it may have been as follows~

Ship P~SLand Sea G

If the ship S at sea is sighted from the top of the cliff T, then by using a special instrument,much like a modern protractor, the angle GTS can be measured. Then point P on the ground is sighted so that LGTP _= LGTS. Then the distance on the ground PG is measured.

Write a formal proof to show that PG = GS, the inaccessible distance along the sea. 22

REFERENCES Bergamini, David, Mathematics (New York: Time-Life Books, 1971 ), pp. 40-41. Bunt, Lucas, Phillip Jones, and Jack Bedient, The Historical Roots Of Elementary Mathematics {New Jersey: Prentice-Hall, Inc., 1976), pp. 69-70.

Cajori, Florian, A History Of ~1athematics (New York: The Macmillan Company, 1929), p. 15.-

Eves, Howard E., An Introduction To The History Of Mathemaotics (New York: Holt, Rinehart and Winston, 1976), pp. 55-56.

Groz!l, Vi vi an Shavv, A )urvey Of ~1athemati cs (New York: Holt, Rinehart and Winston, 1968 , p. 176.

11 Somers, Donald L., 11 Pons Asinorum, Historical Topics For The ~1athe matics Classroom (Washington, D.C.: NCTM, 1969), p. 219. Chapter 4

PYTHAGORAS AND THE PYTHAGOREANS

Although there are many stories·about the life of Pythagoras, little is known for certain. He v1as born about 572 B.C. on the Aegean island of Samos, off the coast of what is now Turkey. Pythagoras may have been a student of Thales, probably traveled to Egypt and Babylonia, and is knovm to have settled in Croton on the coast of southern Italy. There he founded the famous Pythagorean school of mathematics, philosophy, and natural science. The Pythagorean school developed into a cult, complete with secret rites and observances.· The word "mathematics" was originated by the Pythagoreans. To them it meant the four subjects of music, arithmetic (number theory as opposed to the practical art of computing), geometry and astronomy - the famous "quadrivium" of the middle ages. Numbers and their relationships fascinated the Pythagoreans. They believed that there was a mathematical explanation of the universe and everything in it. We are told that it all began when Pythagoras noticed that the ratios between C, F, G, and low C, and between their equivalents in any scale, could be expressed as the ratio of tNo \'/hole numbers. From this he concluded that all harmony, all beauty, and all nature could be expressed by whole numbers or ratios of whole numbers.

23 24

The Pythagoreans developed a number theory which was highly unusual and embodied a large mystical element. For example, they classified the even numbers as feminine and the odd numbers as masculine. One was considered the generator of all numbers and stood for God. Four, a square number, was the Pythagorean symbol for justice (render like for like). They regarded five as the symbol for marriage, since it is the sum of the first feminine number, two, and the first masculine number, three. There were also less under standable associations: six was the number of the soul, seven of understanding and health, eight of love and friendship. Another of the many concepts traceable to the Pythagoreans is the classification of all numbers as perfect, abundant, or deficient. A number is perfect if itequals the sum of its proper divisors. Six is the first perfect number, since 6 = 1 + 2 + 3. Twenty-eight is a perfect number also, since 28 = 1 + 2 + 4 + 7 + 14. A number is abundant if it is less than the sum of its proper divisors. For example, twelve has proper divisors 1, 2, 3, 4, and 6, totaling 16. A number is deficient if it exceeds the sum of its proper divisors.

For example, fourteen has divisors i, 2, and 7, totaiing iO. After trying a few numbers, it is easy for students to see that there are many fewer perfect nu~bers than abundant or deficient numbers. Only a very few perfect numbers are known. In studying the above properties of numbers, the Pythagoreans found that .there were pairs of numbers a and b such that the proper divisors of a add up to b, and conversely, the proper divisors of b add up to a. Such numbers they called amicable, or 25

friendly~ numbers. The smallest pair is 220 and 284. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. Their sum is 284. The proper divisors of 284 are 1, 2, 4, 71, 142, and their sum is 220. The geometrical or physical representation of numbers by points (or pebbles) in a plane and the investigation of their resulting properties was a natural study for the early Pythagoreans. 11 Figurate numbers 11 originated with them. Geometry, to the Pythagorean brother hood, was a geometry of aggregates of points, not of lines as in Euclidean geometry. Pebbles were grouped in certain patterns, which

11 11 11 11 resulted in the classification of numbers as triangular , Square ,

11 11 hexagonal , etc., according to the shape of a group of that 11umber of points.

TRIANGULAR NUMBERS

SQUARE NUMBERS

Each of these classes of numbers can be characterized by a certain series. For example, all triangular numbers can be generated by the series 1 + 2 + 3 + . . . + n; a 11 square numbers by either the series 1 + 3 + 5 + 7 + .•. + 2n- 1, or the sequence 26

12, 22, 32, . . . , n2 . It is easy to demonstrate that to get from one square number, 2 n2, to the next higher square number, (n + 1) , when the number is expressed in figurate form, one must add a row of n units to the top or bottom, another row of n units to a side, and then to com- plet~ the square, a single dot in the corner. This is a concrete 2 2 example of the well known identity, (n + 1 ) = n + 2n+ 1. From 2 2 2 t~is, it is easy to generalize to (a+ b) = a + 2ab + b , and illustrate this by the use of figurate numbers. The first identity is illustrated for n = 3 below.

n n 1 n . D Dn I . ·I 1

(n + 1) 2 = n2 + 2n + 1

Another interesting relationship that can be illustrated with figurate numbers is shown below.

. ' . . . ' ' .' . '

If two successive triangular numbers are added together, the result is always a square number. The best remembered of the Pythagorean teachings is the 27

Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

This theorem was known to the Babylonians, more than a thousand years earlier, but historians think that Pythagoras may have formulated the first proof of the theorem. Some authorities believe that Pythagoras gave a dissection proof suggested by the figure below.

a a a b

b b b a

b a b a -. 28

Proof: Let a, b, c denote the legs and hypotenuse of the given r_ight triangle, and consider the two squares in the accompanying figure, each having a + b as its side. The first square is dissected into six pieces, namely the two squares on the legs and four right triangles congruent to the given triangle. The second square is dissected into five pieces, namely the s~uare on the hypotenuse and again four right triangles congruent to the given triangle. By subtracting equals from equals, it now follows that the square on the hypotenuse is equal to the sum of the squares on the legs. Today there exist several hundred proofs of this theorem. Elisha S. Loomis, in a volume called The Pythagorean Proposition, gives 370 different proofs all classified by types. Closely allied to the Pythagorean theorem is the problem of finding integers a, b, c which can represent the legs and hypoten use of a right triangle. Formulas that generate Pythagorean triples were discovered many centuries ago. The history behind the deriva tion of these formulas is the subject of the next unit, 11 Pythagorean

11 Trip1es • The mystical belief in the pervasiveness of whole numbers and ratios of whole numbers was ironically shaken by the Pythagorean theorem. In a right triangle, measures of the sides cannot always be expressed in terms of a ratio of two whole numbers. If the measures of the legs are 3 and 4, then the hypotenuse is 5. But in an isosceles right triangle with sides each one unit in

1ength, the measur·e of the hypotenuse is /2; and /2- cannot be 29 expressed as the ratio of two whole numbers. The discovery of irrational quantities was disastrous to the Pythagorean philosophy. It was so humiliating and frustrating that an effort was made to keep the discovery secret. According to legend, Hippasus, one of the Pythagoreans, angered the gods and was dro\'med at sea because he revealed to outsiders the secret that /2 is in~ational. This discovery was of great importance to mathe matics and is among the most significant contributions made by the I Pythagoreans.

EXERCISES FOR STUDENTS 1. Pythagoras has been called the inventor of musical science. It may surprise you to discover "that there is a very close relationship between mathemat~cs and music. In fact, until about the sixteenth century, music was considered a branch of mathematics. Prepare a report on the mathematics of music. You may wish to investigate the mathematical laws that govern the tones of different musical instruments like the organ, the violin, or the wind instruments. You might be interested in exploring how mathematics is related to harmony. 2. Figurate numbers originated with the earliest members of the Pythagorean society. .These numbers, considered as the number of dots (or pebbles) in certain geometrical configurations, represent a link between Pythagorean arithmetic and geometry. Prepare a report on figurate numbers.

3. Pt~epare a report on Pythagorean astronomy.

4. Pi~esident Garfield inventedanoriginal proof for the Pythagorean 30

Theorem in 1876 when he was a member of the House of Representatives. Leonardo da Vinci also discovered a proof for the Pythagorean Theorem. Report on either one of their proofs or on one of the others recorded by Elisha S. Loomis in The Pythagorean Proposition. 5. Find the length of the diagonal of a square whose sides are each one mile long.

6. Find the measure of the altitude on the base of an isO'sceles

triangle if the measure of its base is 48 and the measure of a leg is 25. 7. The distance between bases on a square baseball diamond is 90 feet. What is the distance between second base and home plate? 8. If the diagonal of a square is 10 feet long, what is the length of one of its sides?

9. Find the mea·sure of the altitude of an equilateral triangle if the measure of each side is 12. 10. The two legs of a right friangle are congruent and the hypotenuse is 8 inches long. How long are the legs?

11; Prove that the ~is an irrational number. (Hint: Read V.C.

Harris, "On Proofs of the Irrationality of 12," The f~athematics Teacher, 1971, 64: 19-21.) 12. Show that a triangle whose sides measure 2x, x2 - 1, and 2 x + 1 (\'!here x > 1) is a right triangle. 2 2 13. Th e express1ons. ( u2 + v2) , (2uv), and (u - v ) represent measures of the three sides of a right triangle. What are these measures if u = 3 and v = 2?

14. Find a point col~responding to /2 on a number line. (Hint: 31

Construct a right triangle with each leg 1 unit in length.) 15. Find a point corresponding to 13' on a number line. (Hint:

Construct a right triangle with legs 1 and 12' units long.)

16. Find a point corresponding to I~ on a number line.

REFERENCES Bunt, Lucas, Phillip Jones, and Jack Bedient, The Historical Roots Of Elementary Mathematics (New Jersey: Prentice-Hall, Inc., 1976), pp. 71-87.

Eves~ Howard E., An Introduction To The Historical Roots Of Mathe~ matics (NewYork: Holt, Rinehartand Winston, 1976)-,pp, 56-64. Groza, Vivian Shaw, A Survey Of Mathematics (New York: Holt, Rinehart and Winston; 1968}, pp. 97-98.

Harris, V. C., 11 0n Proofs Of The Irrationality Of /2," The Mathe matics Teacher 1971, 64: 19-21. Loomis, Elisha S., The Pythagorean Proposition (Washington, D.C.: National Co~ncil of Teachers of Mathematics, 1968),

Shulte, Albert, ''Pythagorean Mathematics In The Modern Classroom, 11 The Mathematics Teacher 1964, 57: 228-232, Chaptet' 5

PYTHAGOREAN TRIPLES

Closely allied to the Pythagorean theorem is the problem of finding a set of three integers that can represent the leg,s and hypotenuse of a right triangle. If a, b, and c satisfy the 2 2 2 relation a + b = c , (a,b,c) is called a Pythagorean triple. The triple is said to be primitive if a, b, and c have no common factors other than one. Methods for generating Pythagorean triples were known many centuries ago. Ancient Egyptian surveyors (2000 B.C.) apparently laid out right triangles by constructing 3-4-5 triangles with a rope divided into 12 equal parts by 11 knots. If so, their methods arose out of purely empirical considerations. The evidence seems good that the Babylonians had a general and systematic solution for the problem of Pythagorean triples. Plimpton 322, a cuneiform tablet discovered in 1945, lists a series of fifteen sets of Pythagorean triples. The size of some of these triples, for example (960, 799, 1,249) and (18,541, 12,709, 18,541), indicates that these were not obtained by trial and error. Plimpton 322 belongs to the period 1900 B.C. to 1600 B.C. and is now the first recorded example of work in number theory. Tradition is unanimous in ascribing to Pythagoras the independent ciscovery of the right triangle theorem named for him - that

32 33 the square on the hypotenuse of a right triangle is equal to the sum of the squares on the two legs. This theorem was known to the Baby lonians, but the first general proof of the theorem may well have been given by Pythagoras, the famous Greek mathematician, around 500 B.C. Pythagoras, Plato (380 B.C.), and Euclid (300 B.C.) each gave solutions to the problem of constructing Pythagorean triples. Let's see if we can discover the formulas of Pythagoras for our- selves. Three of the most commonly used Pythagorean triples are listed in the table below:

a b c 3 4 5 5 12 13

7 24 25

Verify that these are Pythagorean triples. In other words, do they satisfy the relation a2 + b2 = c2? Notice the patterns in the table above: l. a and c are odd, b is even. 2. The two larger numbers are consecutive. (b + 1 = c) 3. The square of the smallest number equals the sum of the other two 2 numbers. (a = b + c) Ask students to construct a trial triple to fit the pattern with 9 as the choice for a. Solution: a = 9, hence b and c must have a sum of 92 or 81 and a difference of 1. 34

b + (b + 1) = 81 2b + 1 = 81 2b = 80 b = 40 and c = 41

2 2 Verify that 9 + 402 = 41 . ( 81 + 1600 = 1681) So ( 9. 40. 41 ) is a Pythagorean triple. Ask students to construct trial triples to fit the pattern with a= 11 and a= 13. Verify that a2 + b2 = c2. It seems that we can continue in this fashion as long as we please, choosing consecutive odd numbers for a and solving for b and c. Now let's generalize the process: lead students to the formulas:

2 x + 1 c =-~- is odd, a = x, b = 2 where x

These formulas are attributed to Pythagoras. They can be used to generate an infinite number of triples all of which are primitive since b and c differ by 1. What happens if we choose ftn even number for a, say a = 4? The resulting triple will satisfy the Pythagorean Theorem, but to the Pythagoreans number meant whole number. To avoid fractions simply double each number and the triple becomes (8, 15, 17). Ver•ify that (8, 15, 17) is a Pythagorean triple. P'lato (380 B.C.) has been given credit for Pythagorean triple formulas that fo 11 ow directly from those attributed to Pythagoras. 35

These formulas allow for cases where the smallest number of the triple is even.

a = 2x, b=i-1, c = i + 1

Notice that a is always even and b - 2 = c.

Fill in the missing numbers: (6, ?, 10) (10, 24, ?) ., (?, 35, 37) Both Pythagoras and Plato could generate an infinite number of Pythagorean triples with their formulas, however, neither one had found general formulas which would produce all sets of triples. They were restricted to the cases where the two larger numbers differ by only one (Pythagoras) or by only two (Plato). Euclid (300 B.C.) gives a completely general and systematic solution to the ~roblem of constructing Pythagorean triples in Book X of the Elements. His formula, probably known to the Babylonians about 1500 years before, will produce all possible Pythagorean triples. The procedure is as follows:

If p and q are integers with p > q and if

2 q b = 2pq, then (a,b,c) is a Pythagorean triple. 2 For example, if p = 3 and q = 1, p2 q = 8, 2pq = 6, p2 + q2 = 10. The Pythagorean triple is (8, 6, 10). If p and q are relatively prime and are not both odd, the triple will be a primitive one. For example, if p = 5 and q = 2, 36

2 2 2 2 p - q = 21 , 2pq = 20, p + q = 29. ( 21 , 20, 29) is a primitive triple. 2 2 2 Have students verify that a + b = c as follows:

2 a + b2 = (p2 - q2) + (2pq) c = (p2 + q2)2

= p4 2p 2 q 2 + q4 + 4 p 2 q 2 = p4 + 2 p 2 q 2 + q4

= p4 + 2p2q2 + q4

a 2 + b2 = c2

EXERCISES FOR STUDENTS 1. Pythagoras is given credit for the following procedure for obtaining Pythagorean triples. Choose an odd number x, and let 2 X - 1 + 1 a x, b = and c = -'-----:::---'-i = 2 2 a. Show that for x = 7. and x = 15 the procedure gives Pythagorean triples. b. Prove that the procedure always produces Pythagorean triples. 2 2 2 (Hint: Verify that a + b = c .) 2. From any Pythagorean triple {a,b,c) we may derive infinitely many other Pythagorean triples (sa, sb, sc) for any positive integer s. From (3,4,5) obtain four other triples. 3. Can all three numbers of a Pythagorean triple be even? Explain. 4. Show that for any integer x, the three numbers 2x, x2 - 1, and x2 + 1 yield a Pythagorean tr·iple. Plato has been credited 2 2 2 with these formulas. (Hint: Verify that a + b = c .) 37

5. Can the triple (7, 24, 25) be obtained from Plato's formulas? Explain. 6. The Hindus (800 B.C. to 500 B.C.) used a rope to lay out right triangles for the construction of alters. They used the Pythagorean set (15, 36, 39) for this purpose. (15, 36, 39) is derived from what primitive triple? 7. Euclid's procedure for producing all Pythagorean triples is as fo,ll ows: If p and q are integers with p > q and if 2 a= p - l, b = 2pq, c = p2 + q2, then (a,b,c) is a Pythagorean triple. If p and q are relatively prime and are not both odd, the triple will be a primitive one (a, b, c will have no common factors other than one). Use Euclid's formula to produce three sets of primitive Pythagorean triples and verify that they do satisfy the Pythagorean Theorem.

REFERENCES Bunt, Lucas, Phillip Jones, and Jack Bedient, The Historical Roots Of Elementary Mathematics (New Jersey: Prentice-Hall, Inc., 1976), pp. 77-79. Eves, Howard E., An Introduction To The History Of Mathematics (New Yo1~k: Holt,Rinehart and Winston, 1976L P"P."" 35-62.

Jones, PhillipS., 11 Mathematical Miscellanea," The Mathematics Teacher 1950, 43: 352.

Jones, PhillipS., "t·1athematical t~iscellanea," The Mathematics Teacher 1952, 45: 269-272. Jones, PhillipS .• "Miscellanea- Mathematical, Historical, Pedagogical , 11 The Mathematics Teacher 1950, 43: 162-164 . . - - l4ilson, John, "A Heuristic Approach To Pythagorean Triples," The Mathematics Teacher 1969, 62: 357-360. Chapter 6

GEOMETRIC CONSTRUCTIONS

Plato (427- 347 B.C.) is recognized as one of the two great philosophers of classical Greece. He exerted a considerable in fluence on the development of mathematics. Plato's influence was not due to any mathematical discoveries he made but rather to his enthusiastic conviction that the study of mathematics furnished the finest training field for the mind. Because of the logic in mathe matics and the pure attitude of mind which its study fosters, mathe matics was of utmost importance to Plato. An outcome of this attitude was hi~ restriction of the geometer to the use of unmarked straightedge and compass alone. These tools were allowed only be cause they were representations of Euclid's postulates. Euclid (300 B.C.) never used the word "compass" in his Elements. Its first three postulates, however, emphasize the idea that the straightedge and compass were the only tools of pure geometry. The first two of these postulates tell us what we can do with the straightedge. We are permitted to draw as much as may be desired of the straight line determined by two given points ..The third postulate, "to describe a circle with any center and distance," and the way it was used, result in a limitation which is usually expressed by saying that Euclid used a "collapsible compass ... In effect, this compass collapsed as soon as one of its points was

38 39 removed from the paper. For instance, Proposition 2; Book I, shows how, by a simple construction of an equilateral triangle with a given segment as one of the sides, it possible to construct any other given segment from a given point. The collapsible compass and straightedge are thus equivalent in use to the modern compass and ruler. Examples of some basic constructions of plane geometr'y are illustrated. In each case, Euclid's figure requiring a complete circle is contrasted with the modern simplified construction that uses only arcs or parts of ~ircles.

CONSTRUCTION EUCLID's. t~ODERN PROBLEM CONSTRUCTION CONSTRUCTION

Bisect a given line E E segment AB.

A --+--B

F F EF bisects AB EF bisects AB

Bisect a given angle,. LABC.

BT bisects LABC BT bisects LABC 40

Construct an equilateral triangle with given side AB. c c

A~ B

~ABC is equilateral ~ABC is equilateral

Construct a perpendicular from a given point P to a given line AB.

PT j_ AB PT _lAB

EXERCISES FOR STUDENTS

Carry out each construction using your compass as if it were a collapsible compass:

1. Bisect angle ABC. A

2. Divide segment ST into four equal parts.

s ------T 41

3. Construct an angle of 60 degrees.

4. Construct a perpendicular to AB at P.

A p B 5. Construct a line parallel to CD through point M. M •

A------B 6. Construct the altitude to side AB in triangle ABC. c A~B

REFERENCES Eves, Howard and Carroll Newsom, An Introduction To The Foundations And Fundamenta 1 Concepts Of Mathematics ( NewYork: Holt, Rinehart and ~~inston, 1964}, pp. 12-13.

Retz, Merlyn and Meta Keihn, 11 Compass And Straightedge Constructions,~~ Historical Topics For The Mathematics Classroom (Washington, D.C.: National Council of teachers of Mathematics, 1969), pp. 192-196. Chapter 7

THE GEOMETRY OF THE FIXED-COMPASS

A fixed-compass is simply a compass restricted to one and the same opening throughout the entire construction. Constructions are performed \'lith fixed-compass and unmarked straightedge. ·Fixed-compass geometry has a rich history. It is associated with AbO'l-Wefa, an Arab mathematician of the tenth century. It was also a tool of artists of the fifteenth and early sixteenth centuries, notably Albrecht DUrer and Leonardo da Vinci. Then in the middle of the sixteenth century, Italian mathematicians, Tartaglia and Cardano, were involved in a mathematical duel where each challenged the other to perform fixed-compass constructions. The geometry of the fixed-compass can be introduced as a dis covery lesson in any typical geometry class which has studied the elementary geometrical constructions with ordinary compass and straightedge. If asked to isolate a significant similarity in the constructions for bisecting a given line segment or a given angle, some students should notice that a single opening of the compass can be used throughout each construction; in fact, that the size of the opening may be arbitrary. Students may at first feel that too small an opening makes the solution impossible, but a little thought will soon indicate that this is no real hardship. The figure shows segment J\B bisected with a compass opening 1ess than ha 1 f of AB.

42 43

Students should be challenged to try to discover if there are other basic constructions which can be carried out using ~fixed or

11 rusty 11 compass and an unmarked straightedge. Encourage students to work together, to pool ideas, and to consider the use of inter mediate constructions, which can be combined into more complicated constructions. It may be worth~t1hil e to suggest such problems to students as the construction of perpendiculars from points on and off a given line; the construction of a parallel to a given line through a given point; and the division of a given segment into any number of equal parts. More challenging problems can be tackled latter. There are a number of ways to construct a perpendicular to a line from a given point on the line. For example, we may erect the perpendicular at point c on a line AB as fallows: Draw C(r) (the circle with point c as its center and fixed radius r) cutting AB in D. D(r) cuts C(r) in E, and E(r) cuts C{r) in F. E(r) and F(r) meet in G. Then CG is the desired perpendicular. 44

A D B

An alternate method, using the properties of the 30° ~ 60° right triangle, is to cut line ED with E(r), thus giving G without the use of point F.

The parallel line construction is elementary if the given point A is at a distance less than fixed radius from the given line BC. This technique is called the "rhombus method." A(r) cuts BC in D. D(r) cuts BC in E. E(r) and A(r) meet in F. Then AF is parallel to BC.

A •

B __::::;> D c 45

We encounter some difficulty if the given point is at a distance greater than the fixed radius from the given line. However, by com bining two basic constructions, the difficulty is found to be only temporary. Again, A is the given point and BC is the given line. First construct a convenient perpendicular at P on line BC. Name the perpendicular PS. Then drop a perpendicular from the given point, A, to PS. Name the perpendicular AD. Then AD is parallel to BC.

A D

B p c

We just saw that there is no trick to constructing a perpendic ular from a given point to~ given line when the point is at a dis tance less than the fixed radius from the line. In fact, this is one of the constructions that is no different with ordinary compass and straightedge. p

A 46

The more challenging case is, of course, when the distance from the given point to the given line is greater than the fixed radius. But again, by combining two previous constructions, the difficulty is found to be only temporary. Challenge your students to tackle this one on their own in exercise #16.

A B c D

2. Divide segment FT into four equal parts.

F T

.....'l Bisect each angle. X

c 4. Divide angle ABC into four equal angles.

s~------c 47

5. Construct a perpendicular to AB at C. {Hint: Draw C(r) cutting AB in D. D(r) cuts C(r) in E, and E(r) cuts C(r) in F. E(r) and F(r) meet in G. Then CG is the desired perpendicular.)

A c B

6. Construct a perpendicular to AB at c, using the properties of the 30° - 60° right triangle. (Hint: Draw C(r) cutting AB in D. D(r) cuts C(r) in E. Then cut 1i ne ED with E(r) giving point G. CG is the desired perpendicular.)

A. c B

7. Construct perpendiculars to AB at A and at B.

A B

8. Construct a 45 degree angle.

9. Construct a square using a compass opening equal to the side of the square.

10. Construct a tan9ent to circle 0 at point F. 48

11. Construct a perpendicular from P to ED using a compass opening greater than the distance from P to ED. • p

E D

12. Bisect chord AB and it•s arc.

13. Construct the altitude to side CD. Use any convenient fixed compass opening.

14. Construct a line parallel to AB through point P, using a fixed-compass opening greater than the distance from P to AB. (Hint: Use the ••rhombus method.'')

• p

A B

15. Construct a line parallel to AB through point F, using a compass opening less than the distance from F to AB. (Hint: Construct a convenient perpendicular at P on line AB. Name the 49

perpendicular PS. Then drop a perpendicular from the given point, F, to PS.)

A------B

16. Construct a perpendicular from P to AB using a compass open ing less than the distance from P to AB. (Hint: Construct any convenient perpendicular to the given line. Then draw a parallel to this line through point P.)

' p

A------B

REFERENCES Hallerberg, Arthur E., "The Geometry Of The Fixed-Compass," The Mathematics Teacher 1959, ~: 232-237

Retz, Merlyn and Meta Keihn, "Compass And Straightedge Constructions,11 Historical Topics For The t·1athematics Classroom O~ashington, D.C.: National Council of Teachers of Mathematics, 1969), p. 194. Chapter 8

THE FIXED-COMPASS AND REGULAR POLYGON CONSTRUCTIONS

Abu'l-Hefa, an Arab mathematician of the tenth centu~y, was the first to write about the problem of systematic fixed-compass con structions. He discussed five problems in detail which require that the construction is to be accomplished using only one opening of the compass. Each is concerned with the construction of regular poly gons from two starting points: one on a given side and the other inscribed in a given circle. In each case, the fixed-compass open ing is specified as being equal to the length of the given side or the radius of the given circle.

Let's follow Wefa's steps to inscribe a square in a given circl~ using a single opening equal to the radius of the circle. Given a circle with radius r and center S. Draw any diameter AG. A(r) (the circle vii th point A as its center and fixed radius r) gives

Z, and G(r) gives T. Draw AT and GT which meet in M. join and extend MS, cutting the circle in B and D. Then ABGD is the desired square.

50 51

The ingenuity displayed .in l~efa•s regular pentagon construction is seldom surpassed in all later development of fixed-compass geome try. The construction is accomplished using only a single opening of the compass equal to the length of a given side. You may choose to do the construction in class, to make a transparency of the com pleted construction, or to challenge your students to follow Wefa•s step by step solution. Procedure: On given side AB, draw a perpendicular at B with fixed-compass opening r equal to AB. On this perpendicular mark off B{r), giving c. Find D, the midpoint of AB. Join CD. D( t') gives s on CD. Find K, the midpoint of DS. At K erect the perpendicular to CD, cutting AB extended in E. A(r) and E(r) meet in M. Join Bt'l and extend beyond M. M(r) cuts BM extended in z. Triangle ABZ is the 11 triangle of the pentagon ... 52

Z(r) and B(r) meet in H, and Z(r) and A(r) meet in T, so that ABHZT is the desired pentagon .

..______.__ __,__--~- D B E

AbG'l-Wef~ gives no hint in his work as to why he proposed and solved these regular polygon constructions with a single opening of the compass. The most plausible explanation seems to be that fixed-

compass constructions were fir~t developed to answer a practical need for regular polygon constructions in art, architecture, and the construction of scientific instruments. The constructions were more efficient, not because it was difficult to adjust the compass, but simply because additional adjustments might possibly cause some in- accuracies. It appears that interest in fixed-compass geometry began, as in so many other topics in mathematics, in the attempt to 53 find a practical solution to a common problem. The next references to fixed-compass constructions are found at the end of the fifteenth and the beginning of the sixteenth centuries. Again, the use of the fixed-compass as a practical device is rather clearly indicated. It was one of the drawing instruments of the artists and artisans of the day. Both Albrecht DUrer and Leonardo da Vinci described constructions based on just one compass opening in their works. Many of these were related to the construction of regular polygons, useful to the artist in decorations and architec ture. In 1525, Albrecht DUrer wrote a book of instructions on the art of measuring with compass and straightedge. Several regular polygon constructions require use of a fixed-co~pass. The inscription of a regular hexagon in a circle and the construction of a regular penta gon on a given side are two of them. DUrer's construction of a regular hexagon presents no special advancement. His method is ours today. Draw a circle with fixed radius r. Draw any diameter, AB. Starting at point A, mark off arcs equal to the radius of the circle. Then connect the six points. 54 ' '

DUrer's construction of a regular pentagon on a given side with the fixed-compass opening equal to the given side is more easily

executed than AbG'l-Wef~'s construction. However, it is only an approximation. Given segment AB, with AB = r. Draw A(r) and B(r). The circles intersect in C and D. Join CD. D(r) cuts

CD in E. D(r) also cuts A(l") in F and B(r) in G. EF extended cuts B(r) in K. GE extended cuts A(r) in ~- K(r) cuts CD extended in I. Then ABKIH is the pentagon. DUrer was only concerned as to whether a construction was accurate enough for his purposes, as this construction was.

Another approximate construction, given by Albrecht DUrer, is that of a regular nonagon inscribed in a given circle with radius r and center 0. Draw a circle, with radius equal to 3r, concentric to the given circle. Using a fixed-compass opening equal to 3r, divide this larger circle into six equal parts by the points A, B, 55

C, D, E, and F. Then draw B(3r) and F(3r) cutting the given circle in N and M, respectively. Then MN approximates the side of the nonagon. The approximation, however, is really not very good, as appears after the construction has been finished. DUrer uses the fixed-compass as dividers for laying off equal units of length along the circumference of the circle. A second fixed setting is used to complete the construction.

Of greater interest are the contributions of Leonardo da Vinci (1452 - 1519). Leonardo considered the fixed-compass as a convenient tool for the artist and architect and not as a device of theoretical interest to mathematicians. Realizing that there are natural limita tions to the accuracy of all such constructions, he seems only to 56

have been concerned as to whether a construction was accurate enough for his purpose. Proper compass openings were found by a trial method of adjusting. A representative construction of Leonardo is that for dividing a circle into 3, 5, 6, and 30 equal parts. Given a circle with A, B, C; and D points of the inscribed hexagon. Draw AD. D(r) cuts AD in N. Draw BN. Extend BN so that it cuts the circle '"' in, M. Then, Ao = 1/3, At~ = 1/5, fil = l/6, EM = 1/30 of the circumference. Here the sides of the pentagon and hence of the 30- gon are only approximate.

In other constructions Leonardo proposed to divide circles into 3, 7, 8, 9, 18, and 24 equal parts using a fixed-compass. Actually, only the lengths of the sides of the required polygons were found. The compass would then have to be reset to the proper opening and the points stepped off around the circumference. Some of the above lengths are only poor approximations, so much so that 57

Leonardo sometimes wrote "falso" next to the construction. He used freely, without acknowledgment, the common knowledge available to the artist and engineei of that time, and he continued. to experiment for easier ways of obtaining practical results. It appears that fixed-compass constructions were part of a common body of practical geometrical knowledge known and used by the artist and artisan of that day. Analysis of great paintings reveal structures based on line and circle, as illustrated by many of the paintings analyzed by Charles Bouleau in The Painter's Secret Geometry:

Piero di Cosima: Holy Family. The circle is capable of containing other geometrical figures. In this picture the figures are clearly arranged within a square standing on its point. The two trees to the right and left reveal the presence of another square standing on its side. (Bouleau, 1963, p. 38) 58

EXERCISES FOR STUDENTS Carry out each construction with fixed-compass and straightedge according to instructions. Use any convenient radius unless instructed to do otherwise. 1. Inscribe a square in a given circle, 0, using a single compass opening equal to the radius of the circle, as Wefa did. (Hint: Draw diameter AG. A(r) gives Z, and G(r) gives T. AT and ZG meet in M. Join MO, cutting the circle in B and D. Then ABGD is the desired square.) 2. Use the results of exercise #1 above, to inscribe an octagon in a given circle. 3. Inscribe a regular hexagon in a circle. 4. Inscribe an equilateral triangle in a circle. 5. Inscribe a regular dodecagon in a circle. (A dodecagon is a polygon with 12 sides.) 6. DUrer, in his ''Course in the Art of Measurement with Compasses and Ruler, 11 attempts, like medieval artists, to construct the various geometrical figures easily, with the compass. He was perfectly familiar with the classic way of constructing a pentagon, but he proposed another, done with a single angle of the compass. It is a good approximation. The figure below is DUrer's constructio~. Try to reproduce it, using fixed-compass opening equal to the given side of the pentagon as DUrer has done. 59

7. Follow Leonardo•s step by step solution for dividing a circle into 3, 5, 6, and 30 equal parts. Given a circle with A, B, C, and D points of the inscribed hexagon. Draw AD. D(r) cuts AD in N. Draw BN. Extend BN so that it cuts the circle in M. _....., _...... ,...... ~ Then AD = 1/3, AM = 1/5, AC = 1/6, CM = l/30 of the circumference. The sides of the pentagon and hence of the 30-gon are only approximate.

A c

8a. Construct an equilateral triangle on a given line segment, using a compass opening smaller than the given segment. b. Construct an equilateral triangle on a given line segment, using a compass opening larger than the given segment. 60

Historical note: In 1613, Pietro Antonio Cataldi published in Italian the first six books of Euclid "reduced to practice." Cataldi challenges students to tackle the equilateral triangle constructions above. For this problem Cataldi simply gives two drawings and says, "let the figures speak for themselves."

REFERENCES Bouleau, Charles, The Painter's Secret Geometry- A Study Of Com osi tion In Art THew York: Harcourt, Brace and World, Inc., 1963 , p:--38-. -

Eves, Howard W., Mathematical Circles Squared (Boston: Prindle, Weber and Schmidt, Inc., 1972), p. 63. Ha 11 erberg, Arthur E., "The Geometry Of The Fixed-Compass," The Mathematics Teacher 1959, ~: 230-244.

Retz, ~1erlyn and Meta Kei hn, il Compass And Straightedge Constructions ;• Historical Topics For The Mathematics Classroom (Washington, D.C.: National Council of Teachers of Mathematics, 1969), p. 192-196. Chapter 9

ARCHU1EDES

Archimedes is regarded as the greatest mathematician and scientist of the ancient world. He is ranked with Sir Isaac Newton of England (1643- 1727 A.D.) and Carl Freidrich Gauss of Germany (1777- 1855 A.D.), as one of the three greatest mathematicians of all time. Archimedes was born in 287 B.C. in Syracuse, Sicily. He was the son of an astronomer and a kinsman and friend of King Hieron of Syracuse. He liven most of his lif.e there. The amazing genius of Archimedes is seen in his many creative works, ranging from pure geometry to applied mechanics. So power ful was his insight that his original contributions included investi gations into the subject that was some 1800 years later called

11 Calculus... But Archimedes was also a great inventor and scientist. Unlike many Greek mathematicians, he was interested in practical applications. When he was engrossed in mathematics, Archimedes lost all interest in food, rest, or comfort. He would sit for hours drawing geometric figures and studying them to discover new ideas. Archimedes was the first man to determine the formula for the volume of a sphere. He also discovered that the area of a sphere is four times as great as the area of the circle about the diameter.

In his vmrk _On The .?_phere And Cylinder, Archimedes proved a theorem which was his favorite. He demonstrated that the volume of

61 62

a sphere is two thirds the volume of the smallest possible cylinder that will enclose it.

4 3 vsphere = 3 TIR

3 Vcylinder = 2TIR

vsphere 2 vcylinder = 3

Archimedes was so proud of this result that ~e asked that the sphere-and-cylinder diagram be engraved on his tombstone. A variation of his favorite theorem states that the volumes of the (double) cone, the sphere, and the cylinder illustrated below are in the ratio 1:2:3.

2 3 Vcone = - TIR i 3 2R = -4 TIR 3 ! vsphere 3 3 vcylinder = 2TIR

Thus cone:sphere:cylinder = 1:2:3. Archimedes is also believed to have discovered that the area of ' any triangle is determined by the 1 engths of its sides. The area of a triangle with sides a, b, and c and semi-perimeter s

(s -:---2--_ a + b + c) is given by the formula, :•. ... 63

A = I s(s - a)(s - b)(s - c)

This formula is often attributed to Heron of Alexandria, a mathe matician of the first century A.D., who was the first to provide a forma 1 proof. A mathematical method that Archimedes used in many discoveries was to divide a geometric figure into many parts. For example, he calculated the area of a circle by dividing it into smaller and smaller rectangular strips. As he increased the number of strips, he reasoned that the sum of the areas of the rectangles would be closer and closer to the area of the circle. This is very much like the method that we use today in the branch of mathematics known as cal culus. He used the same method to find areas of regions bounded by such curves as parabolas and spirals. Three of Archimedes' extant works are devoted to plane geometry. They are The t1easurement Of fl Circle, Quadrature Of The Parabola, and On Spirals. In The ~1easurement Of A Circle, J1.rchimedes proved that the area of any circle is equal to the area of a right triangle with one leg equal to the radius and the other equal to the circum ference of the circle. You will be challenged to prove this theorem in exercise #5. In this same work, Archimedes made the first scientific attempt at computing n, the ratio of the circumference of a circle to its diameter. We may wonder why Euclid (300 B.C.) and his contempories apparently never calculated n. 64

Archimedes used inscribed and circumscribed regular polygons to determine the value of n. He reasoned that the circumference of a circle lies between the perimeter of any inscribed regular polygon and that of any circumscribed regular polygon. He also reasoned that as the number of sides of the polygon increased, the perimeter of the inscribed (or circumscribed) regular polygon approaches the circumference of the circle as a limit. Archimedes started with regular inscribed and circumscribed hexagons, and continually doubled the number of sides until perimeters of polygons of 12, 24, 48, and

96 sides were obtained. By this method he was able to fix n as lying bet\11een and 3 j~ To two decimal places, TI is given by 3.14. The method inaugurated by Archimedes is known as the

11 Classical method 11 of computing n. Approximations for n can be carried out to as many places as the computational ability and the perseverance of the worker permit. In 1579 the French mathematician 16 Francois Viete used polygons having 6 • (2 ) = 393,216 sides to find TI correct to nine decimal places. In 1610 Ludolph van Ceulen, 62 of Germany, used polygons having 2 sides to compute TI to thirty-five places. In a somewhat similar manner, Archimedes found some remarkable ration a 1 approximat·i on·s to i rrationa 1 square roots. For examp1 e, he calculated 13 by showing that

265 < IT< 1351 153 780 65

Even more remarkable is the fact that Archimedes made all his discoveries without a convenient numeration system. The Greeks used the letters of their alphabet as numerals without place value. This system made calculation quite cumbersome. Archimedes faced up to this severe disadvantage in a scientific treatise known as The Sand- Reckoner. He based his number system on the Greek "myraid", 1t1hich is equal to 10,000. Numbers up to a "myriad of myriads", or 108, were called numbers of the first order. Archimedes then used this

11 myriad of myriads" as the unit of the second order and continued in this fashion to develop an elaborate system of numeration capable of handling numbers as large as desired. Archimedes applied this system to compute the number of grains of sand in the universe at 63 1 ess than 1 o . Archimedes is also famous for many scientific discoveries and mechanical inventions. The best known story of his discoveries involves a basic law of hydraulics. It is said that while taking a bath, Archimedes suddenly realized that an object floating on water displaced its own weight of water and in general that a solid immersed in a liquid loses exactly as much weight as the weight of the liquid it pushes aside. Forgetting to put on his clothes, he jumped from the bath and ran naked through the streets of Syracuse shouting, "Eureka, Eureka!" ("I have found it, I have found it!") This discovery enabled Archimedes to confirm King Hieron's suspicion that a goldsmith had substituted silver for gold in making a crown for him. 66

Archimedes also layed the theoretical foundations for the

m~chanical laws governing levers and pulleys, which are still studied in physics. He is supposed to have discussed the law of the lever in these terms: "Give me a place to stand, and I shall move the earth.'' Writing to King Hieron, he stated that if there were another earth, by going to it, he could move this one. It is said that Archimedes loaded a ship to full capacity with passengers and

frei~ht, and then preceded to draw the ship out of the water by using a simple pulley. The war machines of Archimedes, in which he applied his dis coveries of pulleys, levers, cranes, burning mirrors, and other devices, were chiefly responsible for the delay of the fall of Syracuse to Rome. He is said to have held an invading Roman fleet at bay in Syracuse harbor for three years with devastating catapults and big-beaked, ship-biting iron claws of his own design. It is also reported that he set the Roman fleet on fire, using mirrors to focus the concentrated rays of the sun on the ships. Eventually, the Roman general Marcellus took the city in 212 B.C. As the story goes, Roman soldiers found Archimedes drawing geometric figures in the sand. The shadow of one of the soldiers fell across his diagram, and Archimedes is said to have exclaimed, ''Don't dis turb my circles!" His death by the Roman's sword marked the be ginning of the decline of the intellectual progress of the Greeks. 67

EXERCISES FOR STUDENTS 1. Cicero has related that when serving as Roman quaestor in Sicily

he found and repaired Archimedes• neglected tomb, upon ~tlhich was engraved a sphere inscribed in a cylinder. This figure commemorates Archimedes• favorite work, On The Sphere And Cylinder. Verify the follovJing two results established by Archimedes in this \'/ork: a. The volume of the sphere is two thirds that of the circumscribed cylinder. b. The area of the sphere is two thirds of the total area of the circumscribed cylinder.

In 1965 it \'las r~ported that, \'lhile excavating for the foundations of a hotel in Syracuse, the tomb of Archimedes, which after Cicero•s time had again become neglected and lost for two millennia, was accidentally refound. 2. Find the area of a 3-4-5 right triangle 1 a. by using the formula A = 2 bh. b. by using Archimedes• formula: A = I s(s a)(s b)(s - c) 3. Use Archimedes• formula Bbove to find the areas of the triangles whose sides are the following lengths. Express each answer in simplest radical form. a. 8,15,17.

b. 7, 20, 23

c. 3' 50' 51 4. Suppose a triangle has sides of lengths 4, 6, and 10. a. Try to use Archimedes • formula to find its area. 68 ' .

b. The result is quite peculiar. Can you explain why it turns out as it does? 5. Assume that the area of a circle is nr2 and the circumference is 2nr. Verify Archimedes' theorem: The area of any circle is equal to the area of a right triangle with one leg equal to the radius and the other equal to the circumference of the circle. 6. In the figure, the arcs are semicircles and have AB, BC, and

AC' as diameters; BD _L AC. The figure bounded by the three semi

circles is called an "abelos, .. or a 11 Shoemaker's knife, .. because of

its shape. Archimedes is credited with the discovery of the fo 11 0\"[ ing properties of the .. shoemaker's knife." Try to prove them.

a. The area of the "abelos" is equal to the area of a circular region with diameter BD. b. The length of the common external tangent-segment (EF) is equal to BD. c. A. E, and D are collinear; C, F, and D are collinear. 7. There are several kinds of spirals. The logarithmic spiral was discovered by Descartes, the man who invented coordinate geometry. Archimedes \

8. The Greek name for 10,000 was 11 myriad. 11 Hrite this number, using an exponent.

9. Archimedes began by thinking of a 11 myriad of myriads. 11 ~Jhat did he mean by this? What would we call this number? Write it in expo- nential form.

10. Archimedes called a 11 myriad of myriads 11 an 11 0ctade_.1 Can you explain why? (Hint: What do an octet and an octopus have in common?)

11. Archimedes named the number 1016 11 the second octade. 11 What is our name for this number? Notice that 1016 = (108)2, so that 11 the second octade 11 could also be called 11 an octade squared. 11

12. What number do you suppose Archimedes meant ·by 11 the third oct a de"? L·Jrite it as a power of 10. 13. An extraordjnarily large number thought of by Archimedes was "an octade of octades," which we would write as

1 ~0800 '000 '000

If this number were to be written out in full, what would you have to do?

REFERENCES

Barava 11 e, Hermann von, "The Number n," Hi stori ca 1 Topics For The Mathematics Classroom (Washington D.C.: National Council of Teachers of Mathematics, 1969), pp. 150-151. Bell, E. T., Men Of Mathematics (New York: Simon and Schuster, l93n, pp. 19-·3"4-:-- Bergamini, David, Mathematics (New York: Time-Life Books, 1971), pp. 47-50. 70

Bunt, Lucas, Phillip Jones, and Jack Bedient, The Historical Roots Of Elementary Mathematics (New Jersey: Prentice-Hall, Inc., 1976), pp. 110. Eves, Howard E., An Introduction To The History Of Mathematics {New York: Holt, Rinehart and Winston, 1976), pp. 97, 135-138, 154-155.

Groza, Vivian Shaw, ~ Survey Of Mathematics (New York: Holt, Rinehart and Winston, 1968), pp~ 100-102, 174. Jacobs, Harold R., Mathematics A Human Endeavor (San FrancAsco: W. H. Freeman and Company,-1970), pp. 144-145, 284-291. Rosskopf, Myron, Harry Sitomer, and George Lenchner, Geometry ·(New Jersey: Silver Burdett Company, 1971), p. 437. BIBLIOGRAPHY

Bell, E. T., Men Of Mathematics, New York: Simon and Schuster, 1937. --

Bergamini, David, Mathematics, New York: Time-Life Books, 1971 Bouleau, Charles, The Painter's Secret Geometry- A Study Of Composition In .n.rt, New York: Harcourt, Brace and l~orld, Inc., 1963. Bunt, Lucas, Phillip Jones, and Jack Bedient, The Historical Roots Of Elementary ~1athematics, New Jersey: Prentice-Hall, Inc., 1976. Cajori, Florian, A History Of Mathematics, New York: The Macmillan Company, 1929. Eves, Howard E., An Introduction To The History Of Mathematics, New York: HOlt, Rinehart andWi nston, 1976-.-

----~~·, Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, Inc., 1972. , and Carroll Newsom, An Introduction To The Foundations ----=A=-n_,d=--=Fundamenta 1 Concepts OfMathemat i cs, NewYork: Holt, Rinehart and Winston, 1964.

Groza, Vi vi an Shaw, A Survey Of t~athemat i cs, New York: Holt, Rinehart and Winston, 1968. Hallerberg, Arthur E., "The Geometry Of The Fixed-Compass," The f~athemati cs Teacher 1959, .?1_: 232-237. Harr·is, V. C., "On Proofs Of The Irrationality Of /2," The Mathematics Te.acher 1971, 64: 19-21. Jacobs, Harold R., Mathematics A Human Endeavor, San Francisco: W. H. Freeman and Company,-1970.

Jones, PhillipS., 11 Angular Measure- Enough of Its History To Improve Its Teaching," The Mathematics Teacher 1953, 46: 419-426. -

, "Mathemat i ca 1 Mi see 11 anea, 11 The ~1athemat i cs Teacher 1950,43: 352.

.]1 72

-~-::;-:;~-~ "t1athematical Misceilanea," The Mathematics Teacher 1952, ~: 269-272. , "Miscellanea -·Mathematical, Historical, Pedagogical," --~-~The Mathematics Teacher 1950, 43: 162-164.

Kline, Morris, Mathematics In Hestern Culture, Ne~tt York: Oxford University Press, 1953-.- Loomis, Elisha S., The Pythagorean Proposition, Hashington, D.C.: National Council of Teachers of Mathematics, 1968. National Council of Teachers of Mathematics, Historical Topics For The Mathematics Classroom (thirty-first yearbook), Hashington, D.C.: NCTM, 1969. Ross kopf, t1yron, Harry Si tomer, and George Lenchner, Geometry, New Jersey: Silver Burdett Company, 1971.

Shulte~ Albert, "Pythagorean Mathematics In The Modern Classroom," The r~athemati cs Teacher 1964, 57: 228-232. · Smith, David Eugene, History Of Mathematics, 2 vols., New York: Dover Publications, 1951-.-

Hilson, John, "A. Heuristic Approach To Pythagorean Triples~" The Mathematics Teacher 1969, 62: 357-360.