~J:J'i- LIST of ILLIDSTFJ1.'Rions Nlaps Page Map 1

ENRICHING HIGH-SCHOOL ALGEBRA THROUGH TEE USE OF HISTORICAL MATERIALS

A Thesis Presented for the Degree of Master of Arts

by ,,, ,/ Paul BJI Boeckerman,.. B.s •

. .:: : :......

TBE OHIO STA'Th: UNIVERSITY l 9 4 7

Approved by: ~J:J'I- LIST OF ILLIDSTFJ1.'rIONS NlAPS Page Map 1. Spread of Ancient Babylonian, Egyptian, and Greek Influence. 24

Map 2. The ~xtent of the Known World at the Time of Ptolemy The Lands Conquered by the Barbarian Invaders in Vie stern Europe. The Extent of the East Roman Empire. 25

Map 3. Extent of Moslem Conquest Rise of Universities CrusaO.es 43 Map 4. Caravan and Trade Routes of the Micidle Ages Teaching Centers of Hindu Mathematicians 44 PIC·ruHES Numerals of the 28th Century B.C., A Sumerian Tablet. 77 E6yptian Hieroglyphic Notation. 81 A Problem in Hieratic from the Ahmes Papyrus and its Equivalent liie roglyphic Form. 82 ·rhe Ancient Maya Calendar-Script. 91 Two Sixteenth Century Modes of Reckoning. 94 Counter Reckoning in 1514. 94 Early Forms o:f the Hindu-Arabic Numerals. 97

Egyptian Word Proble~. 130 Symbols for Roots of Algebraic Polynomials. 131 Symbols for the Representation of Unknown Quantities. 132 The Sign of Equality 132 Variants of the Radical Sign. 133 Methods of Indicating Plus and Minua 133 Map of the Worlu by Hecataeus, 517 B.C. 139 Map of the World according to Eratosthenes. 139

Ptolemy's Map of the World. 140 Page Chapter I

Int r·oO.uct ion • 1

Statement of the Problem 7

hel&tea Stuc:i.ies 7

Hesolution of the Problem 8

Foot-Notes 9 Chapter II Chronolo6ical Outline

Introduction 11 First Period c. 50,000 B.C. to c. 500 A.D. 16 Second Period c. 500 A.D. to c. 1600 A.D. 33 Third Period. c. 1600 A.D. to 1700 A.D. 51

Foot-Notes 63 Chapter III Extension of the Number System

Introduction 68 The Number Concept in Pre-History 70 The Natural Numbers Babylonian Number System 75 Egyptian Number System 79 Greek Number System 83 Roman Number System 86 Mayan Number Systerr: 90 Hindu-Arabic Number System 92 Extensions to the Natural Numbers 101 Fractions 102 Irrationals 110 Negative Numbers 113 Complex Numbers 117 Foot-Notes 121 Pae;e Chapter IV Sug'-:;estions for tbe Use of Historical Materials in the Classroom 124

Foot-Motes 150 Chapter V

Surrnnary 153 Sug";estions for :b,urther Study 156

Bibliography 158 CHAPTER I

INTRODUCTION

There is little doubt that mathematics can be taught, and taught well, without any reference being made to its historical development, or to its in£luence in the develop ment of civilization. Such a method may, perhaps, have certain definite advantages. Mathematica is an abstract science, more abstract today in its t·oundations than ever before, and, as such, may be studied independently of any physical reality. But to teach it without occasional refe:P.. ence, at least, to its historical development is to with hold from the student an enrichment of understanding that only the history of the subject can give. Of what value is the history of mathematics to the teacher of mathematics? In addition to information a well- informed cultured person should know, the history of mathe matics provides the teacher with a perspective that enables him to discriminate between the essentials and non-essen- 1 tials in his teaching field. History shows how the mathematics of the Greeks, though sufficient for the builders of their civilization, is wholly inadequate for the builders of present-day civilization. The arithmetic of the Greeks has long ago been discarded for better methods. Advances and refinements are characteristic of living mathematics, but there has always been a time lag between an advance and its incorporation into the educational program of the age. Statistics has been an important and relatively recent mathe- - 2 -

matical development, It is widely used today in many differ ent fields, ranging from Gallup polls to nuclear physics. But how many secondary schools, at present, give students a working knowledge of statistics? Some authorities, even, are of the opinion that if all obsolete material were dis carded from high-school mathematics there would be sufficient time to give the f'undamentals of calculus, already in the secondary schools. Even a reading knowledge of what modern

scientists imagine the world to be requires a working know- 2 ledge of the elements of the calculus. As mathematics and the teaching of mathematics advance it is the well-informed and alert teacher who compensates for the time lag in textbook material. There is yet another important reason why teachers of mathematics should be familiar with the history of their subjects. The inherent difficulties of many topics of mathe- 3 matics may be seen in their historical backgrounds. "History," says 9ajori, "emphasizes the importance of giving graphical representation to negative numbers in teaching algebra. Omlt all illustrations by lines, or by the thermometer, and negative numbers will be as absurd to modern students as they 4 were to the early algebraist a. 11 Negative numbers were 1 absurd' or 1 fictitious 1 to early writers until Fibonacci, in the early thirteenth century, suggested a meaning for them by interpreting a negative number in a money problem as a 5 loss instead of a gain, and until Descartes, in the seven-

teenth century, gave them a geometrical interpretation in - 3 -

his analytic geometry. It may be important to point out that negative numbers today are defined in a manner independent of any geometrical interpretation. Of what value is the history of mathematics to the students? It is quite easy to see that, for a student with a natural inclination for the study of civilization and its developnent, the story of mathematics may be a fruitful ap proach to a much wider study of mankind. All students, on the other hand, may deepen their appreciation of the value of such attitudes as tolerance and cooperation by underatand- 7 ing the nature of mathematical growth. Mathematica is rarely conscious of nationalistic, racial, or religious prejudice. The Arabs eagerly sought after Greek and Hindu mathematics. Thia learning the Moslema later passed to Christian Europe. The personalities of mathematicians in the eighteenth century took on an international character, as these mathematicians lectured and carried on research at the various royal courts and universities of ~urope. The development of mathematics, both pure and applied, moreover, has largely been cooperative in nature, the product of the activity of many men, united by a common interest. Eighteen hundred years after Apollonius wrote his treatise on conics Kepler read the work and applied it. Kepler discovered that his calculations, made from the data so painstakingly collected by Brahe, led to the conclusion that the bodies in the soler system revolved about the sun in elliptical orbits. - 4 -

The history of mathematics as an aid to teaching is a point we cannot af'f'ord to overlook. In all newspaper re- porting, pictorial or written, it is the human interest aspect of an event that is stressed. Why not give the hum.an interest side of mathematics, those numerous episodes and interesting achievements in the lives of mathematicians? Fourier was a great mathematician and a great teacher, and it is said of him that he "enlivened his lectures on mathematics by out-of'-the-way historical allusions (many of which 8 he was the fir·st to trace to their source)". J. V. Collins has developed an interesting variation of the idea of intro- ducing human-interest stories into courses. There are in- tervals of relaxation in all labor, and as a relief during the progress of the teaching period he suggests relating anecdotes, which are mathematical in nature, from the lives of such prominent men as the students meet in their study 9 of literature, history, and science. The human-interest side of mathematics may stimulate student interest. The histories of special topics, on the other hand, when skill- fully woven into teaching procedures, may make the learning of these topics easier, through an understanding of how they developed. The Committee on the function of mathematics in general education in their report, Mathematics in General Education, recognizes the values in an understanding of the history of 10 mathematics. The Committee points out that the history of mathematics, in aGaition to giving new light and life to - 5 -

the subject, leads to an appreciation of mathematics as one of the main instruments by means of which man has construe- ted for him.self the world in which he now lives. The Committee, likewise, points out common misconceptions as to the nature of mathematics and its place in the history of civilization: 1) Mathematics always had its present form and few additions will ever be made. 2) Mathematics is the product of steady inexorable progress from primitive prehistoric fonns of counting and space perception to present com plex techniques. 3) Mathematics is a system of proposition and re lations without relation to personalities and human affairs. 4) Political and universal histories ignore the in1'luence of mat;hematics in historical trends. To correct such misconceptions and to attain the positive values, the Committee proposed that the teacher help students gain understandings of the following sort: 1) Understandings related to the development of mathematics in response to the needs of the times. a) Mathematics as a direct response to the spirit and needs of the times. b) The insuf'f'iciency of the needs and the spirit of the times as an explanation of all mathematical development. 2) Unaerstanding related to the development of mathematics as a science. a) Cumulative character of mathematical develop ment. b) Some non-cumulative aspects of mathematical development. c) Light thrown on the nature of mathematics by certain aspects of its developments. Although the Committee enlarged upon these points, it 6

is not necessary for us to go into their detailed development here. It is important, however, to note the emphasis given these ideas by the members of' this group.

To what extent has the history of mathem~tics been em ployed in teaching high-school mathematics? C. L. McKee published in 1937 in the Mathematics Teacher the results of a small survey he had made of historical materials present 11 in secondary-school textbooks. He uncovered very little material of such a nature. Ee consulted books on methods of' teaching mathematics and again found a comparatively scanty amount of pertinent material. He sent a questionnaire to prominent educators in teachers colleges and to prominent textbook writers, asking for their views on the advisability of training prospective teachers of' ma.them&tics in the use of historical materials. To this he received a unanimous answer in the affinnative. Raleigh Schorling points out probable reasons for the poor use made of historical materi- 12 ala. He mentions that, although there are several excellent histories available, only a very small amount of this material hal'J been adapted to the needs of primary and secon dary school students. Perhaps the most important reason of all, he feels, is the fact that :few teachers are sufficiently acquainted with these materials to use them effectively. Although mathematics, in its traditional treatment, seems to have been taught with little reference to its historical development, we, as teachers, should make serious attempts to remeay this situation. The work of this thesis - 7 -

has been directed towards attaining this end, to some extent at least, in the teaching of elementary algebra.

STATEMENT OF 'rlrE PROBlliM The problem of this thesis is to illustrate methods and procedures that may be employed in making effective use of historical materials in teaching elementary algebra.

RELATED STUDIES A certain amount of work in this field has already been accomplished. E. R. Baker, in his master's thesis, develoi:ed a ready source of' historical material directly related to 13 the concept of measurement. I. H. Brune made a special study of the great Greek mathematicians and of their influ- 14 ence on their own and future civilizations. R. A. Shaw in her master's thesis included the selection of a wide ra:rge of historical materials to be used in the teaching of General Mathematics, Algebra, and Plane Geometry, together with sug gestions for ways of utilizing these materials in providing 15 pupil experiences. To these studies may be added the History and Significance of Certain Standard Problems in Algebra, a short book by Vera Sanford which shows the re lation between the subject matter treated in word problems and the business, economic, scientific, military, and social· conditions upon which these problems are based. Vera Sanford also recognized the need of a history of mathematics adapted to high-school level, and published A Short History of Mathe matics. Tobias Da.ntzig in his work, Number the Language of - 8 -

Science, unfol6.s the story of' numbers as "a histor•ical pa- geant of ideas, linked with the men who created these ideas 17 and with the epochs which produced the men". The problem of this thesis, though it may overlap slightly some of' the above studies, is held to the limits of elementary algebra, the algebra that is ordinarily taught in the f'irat year of high school.

RESOLUTION OF T.Iili PROBLEM In considering the solution of' the problem the author recognized, first of all, the need for an integrated pic ture of the development of elementary algebra in its re lation to the development of mathematics in general and of civilization as a whole, and developed a chronological outline to give this over-all view. Secondly, the author re cognized the usefulness of taking this outline as a point of' departure for developing, in more detail, special topics of elementary algebra that run like threads through its his tory and selected the extension of the number system for such a historical development. The final step in the solution of the problem consists in suggesting ways to uti lize in the classroom the materials developed. - 9 -

FOOT-NO'I'ES FOR CHAPTER I

1. L. C. Karpinski, The History of Arithmetic, p. v. 2. E. T. Bell, "The Meaning of Mathematics," The Eleventh Yearbook, The National Council of Teachers of Mathe Matics, 1936. p. 157. 3. Raleigh Schorling, The Teaching of Mathematics, p. 213, 4. Florian Cajori, A History of Elementary Mathematica, P• 411. 5. E. T. Be11, The Development of Mathematics, p. 157.

6. ~., P• 165. 7. Mathematics in General Education, p. 51. 8. E. T. Bell, Men of Mathematics, P• 193. 9. J. v. Collins, "Correlation of Mathematica with Biography, History, and Literature," School Science and Mathematics, V (November, 1905), p. 640. 10. Mathematics in General Education, "Chapter XI, 11 pp.241-6 6. 11. C. L. McKee, "Historical Material in Secondary Mathe matics," School Science and Mathematica, XX.XVII, (May, 1937), PP• 423-30.

12. Schorling, op. cit.~ pp. 213-14. 13. E. R. Baker, "The Use of Historical Materials in Teaching Measurement," Unpublished Master's thesis, Ohio State University, 1946. 14. I. H. Brune, "Greek Mathematics and Modern Teaching," Unpublished Master's thesis, Ohio State University, 1939. 15. R. A. Shaw, "Historical Materials and Their Use in the Teaching of Secondary II.a.thematics,'' Unpublished Master's thesis, Ohio State University, 1940. 16. Vera Sanford, History and Significance of Certain Stan dard Problems in Algebra. 17. Tobias Da.ntzig, Number the Language of Science, p. vii. CHAPTER II

CHRONOLOGICAL OUTLINE OF TEE DEVELOPMENT OF ELEMENTARY ALGEBRA IN ITS RELA'rION TO TEE DEVELOPMENT OF MATllliMATICS IN GENEF.AL AND TO TEE DEVELOPMENT OF CIVILIZATION AS A WHOLE. INTRODUCTION

The purpose of this chapter is to give an integrated picture of the development of elementary algebra in its re lation to the development of mathematics in general and of civilization as a whole. The history of mathematics is an assemblage of related parts, a complicated assemblage of many intangible parts- call them facts and their interpretations. The mathematical facts, however, form only a part of the whole picture we wish to give. Mathematics is the invention of man, and its development is tied up with the forces that affect man,• Mathematical contr·ibutions, of an individual or of a period, for fu.11 understanding and appreciation, should be viewed, not alone, but in relation to the history of mankind as a whole. Seated on a mountain side with an unobstructed view, one may at a single glance encompass miles of the valley be low. We see villages and hamlets, together with the commu nication systems that unite them--telephone lines, roads, streams. The higher we ascend the greater the territory the eye can take in. We sacrifice detail for breadth of view. To study the beauty of a town in the valley below we must descend. To study the beauty of the houses in the town we must descend to the houses themselves. We sacrifice breadth of view for detail. A chronological outline of the kind we have attempted gives a mountain-top view. We see how small - 12 -

patches of algebra fit into the more extensive patterns of mathematics, and how the patterns of mathematics fit into the panorama of mankind's history as a whole. We can trace the methods by which mathematics spread across the broad plain of' history. The limitations oi· the outline are ob vious: detail is sacrificed for breadth of view. Later we may descend to investigate the details at closer hand. The chronological outline is divided into four columns with the following headings:

l 2 3 4

DATES a a g ven date or at the t time of a given event.)

The advantage of such an arrangement is that, starting from the fourth column and proceeding to the left, one may place a special development in algebra in relation to both mathematics and to important events in the history of ci vilization. Staying in the fourth column, and working upward and downward from a special development, one can see its relation to the development of algebra as a whole. The outline is limited to the indication of basic de- velopmenta. The addition of details might be both desirable and profitable, if this outline could be spread out to form a large wall chart; but to keep the outline within the li- 13

mitations set by the process of binding this thesis, many details had to be excluded. The problem in this thesis is, likewise, limited to the historical mate~ials enriching elementary algebra. Elemen- tary algebra, therefore, sets the limits for the whole stu dy. By elementary algebra we mean the algebra that is or dinarily taught in the first year of high school. As the algebraic content of modern secondary-school texts was practically all in use before the eighteenth century, we will terminate our study of the history of algebra, the history of mathematics, and the history of mankind as a whole with the close of the seventeenth century, except for a few later developments which have influenced the subject. An outline, once it is made, may serve as an overview of a study, a guide for directed read:lngs, or a framework of basic ideas requiring further development. The following outline is the end product of a personal study, but it may now serve any of these three purposes, in addition to giving an integrated picture of the development of mathematics to one who has completed his studies in the subject. To facilitate the work of drawing up this outline it has been divided into three periods of unequal time inter- vals. First Period •••••••••• c. 50,000 B.C. to c. 500 A.D. This first period covers the mathematics of the early Babylonians and Egyptians, rising to a height among the early Greeks, ana declining to a state of dormancy in the Roman Empire and the barbarian invasion of Europe. - 14 -

Second Period •••••••••• c. 500 A.D. to c. 1600 A.D. The second period follows mathematics through the centuries of transmission, when little was added by the Moslem and Arab translators of Greek and Hindu works, to its slow awakening in Eastern Europe in the fourteenth, fifteenth, and six teenth centuries. Third Period •••••••••• c. 1600 A.D. to c. 1700 A.D. The third period finds Europe fUl.ly awake, de veloping the beginnings of modern mathematics. With each period are given brief amplifications of certainjl;ems that appear in the outline, specifically or by implication, and which are of such a nature that they require, for the sake of cla1·ity, more amplification than the economy of' space in the outline permits. Forces or conditions that influenced the development of mathematics (or vice versa) over a great period of time, such as the geographical and clim.atical conditions of ancient Babylon, the effects of the barbarian invasions, the Crusades, the importance of' printing from movable type, the effects of the calculus on civilization, etc., have been amplified briefly in the form of "paragraph" outlines. General mathematical trends, extending over centuries, are best treated in a similar manner. It may be important to point out that, though these paragraphs may appear to give an overview of a period, they fall far short of accomplishing such a purpose. Their primary purpose is to function as an outline, as a complement to the main outline, and, as such, serve the same purposes as the main outline. These purposes we have already ment:kmed. - 15 -

Where possible maps have been added to show graphi cally the extent of the spread of a culture, and the means by which the exchange o1' ideas among people was affected. - 16 -

First Period •••••••••• c. 50,000 B.C. to c. 500 A.D.

CON'rRIBUTIONS OF ANCIENT BABYLONIANS, EGYPTIANS, AND GREI!.KS.

Before passing to the chronological outline for this period, we shall give a brief treatment of' those phases of Ancient Babylonian, Egyptian, and Greek civilizations that were related to developments in mathematics, and a brief treatment of the mathematical trends among these ancient cultures.

ANCIENT BABYLON For our purposes Babylonia shall include all those Semitic Peoples who descended in Prehistoric times from the Southern Grasslands and who settled aaong the Phoenician coast, Asia Minor, and the fertile plains between the Tigris and Euphrates Rivers. For convenience we shall also include the non-Semitic Sumerians, who descended to the head of the 1 Persian Gulf from the mountain regions in the east. The extent of Babylonian and Egyptian culture is indiaated on

Map 1, page 2 4. The plains between the Tigris and Euphrates rivers were so fertile that here, according to existent records, the earliest civilization flourished. From the alluvial 4eposits came the mud for bricks, bricks for building and bricks for writing. Cuneiform writings uncovered since 1929 tell us of an agricultural civilization, requiring a reliable calendar to record the relationships between seasons and the signs of the heavens. - 17 -

Lying in the path of ancient trade routes, Babylonians developed commercial traffic with neighboring and distant tribes. The Babylonians were careful bookkeepers and their detailed records give us a somewhat comprehensive idea of the economic conditions in ancient Babylon. Among the Ba bylonians were such specialists as carpenters, masons, smiths, scientists, mathematicians, poets, _and musicians. The inundations of the rivers were a constant source of danger,--floods, swamps, miasma, and insect scourges,-- until systems of irrigation and appropriate building de- 2 signs gave security.

3 ANCIENT EGYPT Another region, geographically and climatically ideal for permanent habitation as early as the Stone Age, was the land bordering the Nile river. Here the prehistoric nomads of the Egyptian race settled down to living in permanent encampments, which in time grew into cities. The cities u nited into the Lower and Upper Kingdoms, and later these two formed the united Egyptian nation. Egypt's climate-- without storms, cold, f'og, and excessive rain--contributed to the preservation of their accomplishments. Our knowledge of early Egypt is gleaned from their inscriptions and carv- ings on tombs and temples, and from their artifacts and papyri. The Nile was all important to the Egyptians. Its in undations left fertile soil along both banks, an excellent - 18 -

soil for agriculture. It was a great waterway for commu nication and commerce. In addition to the Nile, Egypt commanded outlets and inlets for foreign trade by both Me- 4 diterranean and Arabian seas. The quarries of Egypt were both numerous and of ex cellent quality, while abundant slave labor made possible the back-breaking working of the stone. About 2900 B.C. the Egyptians erected the great pyramid at Gizeh. They used over two million blocks of stone, the largest weigh ing 54 tons, taken from a quarry 600 miles away and raised

200 feet above ground in the pyramid. In 1500 B.C. they set up at Thebes their largest extant obelisk, 105 feet

long, weighing 430 tons. The Egyptians also built sun dials, gnomons, water clocks, ships, aqueducts, and tunnels for providing water 5 supplies. 6 ANCIENT GREECE Before the dawn of written history strange tribes came down out of the north through the Balkans into Greece and mixed with the early inhabitants. These early people were in touch with the.then bronze age civilization on the island of Crete, from which civilization is thought to have 7 come to Greece proper. Greece, with its many natural harbors, quite easily grew into a sea power and became the mother country for the

colonization of the Mediterranean shores. (See Map 1, page - 19 -

24, for the extent of Greek influence.) Grecian hills held unrivalled marble from which were quarried the stone f'or her buildings and monuments. With a climate alternating between invigorating cold and relaxing heat, with a social order favoring an intel lectual leisure class, Greece produced scholars of the first magnitude in philosophy and in the deductive side of mathematics. The mundane aspects of mathematics,--the numerical calculations practiced in trade, commerce, and the sciences, --did not hold the respect of this scholarly class and were left to the merchants and educated slaves. As the fame of the Greek schools spread, more and more scholars flocked to 8 Grecian cities and intellectual activity reached a peak •

• • •

Following the first faint rivulets, from the begin nings of history to 500 A.D., that since have grown into a 9 powerfUl stream of mathematics, do we realize that we have scanned 40,000 of the 55,000 years of history? And for this great bulk of time how small was the sum total of accumulated knowledge, compared to the vast accumulation we have todayl After 40,000 years the curve, that describes mathe matical growth as a function of time, had only begun to 10 show a marked upward rise. And it was the Greeks with their idea of proof that produced this. The peoples included in the term Babylonia (later As syria ruled over these nations) in time brought the a.rith- - 20 -

metic of commerce to a high standard and extended the early work in astronomy. After Assyria fell prey to Persia in 538 B.c., Babylonian mathematics practically ceased to 11 exist. The Egyptians, despite all their splendid buildings and monuments, invented practically nothing, but used, in stead, the invention of others. They had an amazing talent for organization and a sufficient supply of slaves to do the Pharaohs' bidding. What they lacked in mechanical devices they made up for in man power; they had, for example, ropes and sleds to haul their rocks, but no pulleys to lighten the task. The early Babylonians and Egyptians spent unrecorded centuries observing the heavens and accumulating data from which they developed a rhetorical and empirical algebra. Their algebra was not algebra at all in the sense we view algebra today. It was only in the third decade of the nineteenth century A.D. that serious attempts were made to pro- 12 vide a postulational foundation for elementary algebra--a foundation the Greeks gave to geometry. The early members of our human race developed their knowledge within the limitations set by their respective civilizations. In like manner we have set the limitations of our own advancement, and future generations shall probably view us as we view 13 these early peoples. Did the Babylonians have any idea as to the nature and necessity of proof in mathematics? We do not know for sure. - 21 -

Existent records do not indicate the method by which they reached their conclusions. The reason for this withholding of the whole story is quite readily understood when we rea lize that, in those times, the priests were the sacred guardians of knowledge, and they guarded their secrets well. In Egypt the priests of the temples had, likewise, the task of guarding the knowledge of the nation. To the Greeks must be giventhe credit for breaking this tradition. Knowledge in Greece was for anyone who sought it. As the early nations grew and expanded, the sacred knowledge of the calendar, and the mathematics that grew from it, gradually slip ped away from the priests as their special prerogative; but the priests gave way slowly and reluctantly, fearing to lose their power. Until it is shown f'rom existent records, or records. still to be uncovered, that the Babylonians and Egyptians understood mathematics to be a deductive science, the grea test tribute than can be paid them is that they prepared the way for the golden age of Greek mathematics. And what mathematics crystallized from the golden age of Greece? These two supreme achievements: 1. The explicit recognition that proof by deductive reasoning off'ers a foundation for the structure of number and form. 2. The daring conjecture that nature can be understood by human beings through mathematics. 14 Their work on the mysticism of numbers, which they picked up from the East and developed was an unfortunate - 22 -

departure on their part, from rational mathematics to the superstition of numerology. Another black mark against the Greeks was their failure to recognize the worth of algebra. They ignored it completely until the time of Archimedes and Diophantus. • • •

Although the main burden of this chapter is to follow the development of elementary algebra in its relation to the development of mathematics and of civilization as a whole, an additional point, however, is likewise stressed. That point is the means by which the exchange of ideas among people was effected. In ancient times the two main channels of transmission of knowledge were war and trade. The mathematics developed by the early Egyptians un doubtedly spread, to a certain degree, to neighboring coun tries as these lands became satellite states under the force of Egyptian arms. Egyptian officials took up residence in conquered lands to gover·n them and to exact tribute, while many of the conquered were brought to Egypt as slaves. The Israelites, in the course of their history, had the mis fortune to fall prey to both Babylon and Egypt, and the Bible records how difficult it was to eradicate from the Chosen People, once they were free again, the pagan culture they had acquire6. during captivity. We shall assume, then, that some knowledge of the mathematics of a culture became known as far as the influence of that culture extended. - 23 -

The early civilizations carried on a certain amount of commerce with other nations, both by caravan routes and by sea routes. The early Phoenicians were well known as sea farers and traders. Every Phoenician colony on the North African and East Iberian coast became a small canter of Phoenician culture. The Phoenicians undoubtealy left an imprint of their commercial arithmetic at the many foreign ports with which they traded. The following maps show the extent of ancient Babylon ian, Egyptian, and Greek influence; the ancient trade routes; the extent of the known world at the time of Ptolenw, c. 150 A.D.; the lands conquered by the barbarian invaders in western Europe; and the extent of the East Roman Empire, which was unmolested by the invaders. - 24 -

MAP l

SPBEAD Olt' ANCTuNT BABYLONIAN, EGYP·rIAN, AND

GREEK INFLUENCE 15

Extent of Ancient Egyptian and Babylonian influence, c . 3000 B. C. -600 B. C. Extent of Greek influence , c . 600 B. C.-300 B. C. Phoenicia, and Phoenician colonies. Greek trade routes . ' ___ ,.. ____ _ Phoenician trade routes .

+++++++-+ Conquests of Al exander the Great, c . 333 B. C. -2:+11- - 25 -

MAP 2

1. THE b:X'l1Bi.\l'r OF 'rl{J:!. .KUOWN WORLD ( IN BLUE} 16 AT 'filli TIME OF p·rOIBMY, c. 150 A. D.

2. THE LANDS CONQU.b;RED BY ·rHE BARBARIAN 17 INVAlERS IN W.B;STERN EUROPE.

3. THE EX'rENT OF rHE EAST ROMAN EMPIIili!,

WHICH WAS UNMOLESTED BY ·rHE INVADERS. -2. S-f/- - 26 -

CHRONOLOGICAL OUTLINE OF FIRST P~RIOD •••••• c. 50,000 B.C. to c. 500 A.D.

The statements in the following outline have been culled from several sources, and it is beat, perhaps, to iist the sources at this time. The dates and events of universal l:istory may be f'ound in any standard text. Not all a.utho- rities are in agreement as to the exact dates of certain events and to the proper spelling of certain names. It may be well to keep this fact in mind if the reader should wish to confirm or investigate further any of the statements in the outline. In the case of ancient mathematics, books re- cently published may even contain information refuting statements found in earlier histories. The reason for this is that archeological findings in recent years have led to re evaluation of the mathematical productions of the early Ba- 18 bylonian and Egyptian cultures. No doubt, as the work of the archeologista continues, further re-evaluations shall be made, not only of the mathematics of these two early civilizations, but of the early Hindu and Oriental as well. The sources consulted in drawing up this outline were: 1. R. C. Archibal:i, Outline of the History of Mathematica. 2. E. T. Bell, The Development of Mathematics. 3. , Men of Ma.thematics. 4. F. Ca.jori, A History of Mathematics. 5. I. B. Hart, Nl.B.kers of Science. 6. Lancelot Hogben, Mathematics for the Millions. - 27 -

7. L. C. Karpinski, The History of Arithmetic. 8. Vera Sanford, A Short History of Mathematica. 9. D. E. Smith, History of Mathematics, Vol. I, II. DA'l'm> EVENTS MA.THEMATICS PRODUCED EBRA PRODUCED B.C. 50,000 Early Stone Age •••first record1. Means ot making tire disc.

15,000 Kiddle Stone Age •••work1 at art. Drawings show a power ot line, tona, and perspective. u,ooo Nomad• in Egypt.

8000 Egyptian• in permanent ence..mpmenta

6700 Beginnings ot Babylonian calendar. The calendar presupposeai 1) a developed nUllber systems 1exageai mal 1y1tem. Numbers led to the superstition of numerology. 2) a primitive astronomy. Aatronomy led to the superatition ot astrology. 5000 Egyptian nomada well aettleda; CultiTation and storage ot crops Doaestioation ot animal1. _4400 r-4000 D11oovery of metal on Sinai pen I in1ula. , Beginning• of E.gYPtian calendar. The Egyptian calendar re1ulted from a high development in computation and a1tronomy. Egyptian years twelve 30-day months, five days for festivities. Egyptians recognized, but could not correct tor leap years. · Altronom:i E&YI>tians saw the relation ship between the heliaoa.l rising of Sirius and .the annual date at which the lile inundated. 3800 ~tian1 decorated pottery, cloth

Egyptian• freely handled large numbera1 Record of booty on royal mace. Decimal system established.

3200 Upper and Lower Egypt organized and warring.

3000 Birth of Egypt as a united nation. Rapid development in engineering, geom- First written Egyptian records. etry, and close accuracy in :m.ensura- Earliest stone masonry; ezramids. tion of pyramids and nilometers.

Bab lonian written record•. Notable engineering works1 irrigation, building, •urveying. Familiarity with weights, measures, bills, receipts, interest, notes, accounts.

2560 IV ~yptian Dynasty: Wars in Egyptian arithmetic of taxes and oenicia, Western Aaia, Sudan. receip ta, interest rates, barter. First expedition to Punt.

2 DATES EVENTS KATimiiATICS PRODUCED ALGEBRA PRODUCED B.C • ...2466 2250 Khammurabi'• "Code ot Law•"• Ruin• ot oldest known school. Domination ot Babylon eat~ 22 .Areas ot rectangle, right tri Systems of 1imultaneoua angle, cirole. equations. Pythagorean theorem tor spec. Solution of special quad caaes. ratic, sol. of spec. Special volumes1 special pro cubic equations by means portiona. of tables. Tables of reciprocals reduced Possibility of exponential di'r11ion to multiplication. equations tor compound interest problems. Negative numbers appear in connection with simul taneous equations •

.Amenemhat III ot Egypt. Extensive system of irrigation Mosoow Papyrusi contains 25 led to increased knowledge problems. of leveling, surTeying, men suration. Egyptians found the volume of a truncated square pyramid, using the formula, l/3h(a2 + b2 + ab), arrived at empirically. Postal service in Asia Minor Business arithmetic for com under Egyptian control. merce. Census li1t1. Arithmetic and geometric aerie •

1650 Rhind Papyrus copied by Ahmes, Table of unit tractions ot 2 Many of the Rhind problems contains 86 problema. divided by all the odd xwm. we would solve by linear ber1 troa 6 to 101. equations. Egyptian• PrQblezu in 4l_n:!!2n l'!duced solved tllem ~ · trial or to multiplio&tiim.. .false poli ti.on. In geometric problem.a pi : 3.1605. Elementary series. No new ma.thematics until 600 600 B.C.

1500 Er_yptian sundial--oldes~ extant. Second expedition to Punt.

1318 Re.meses II ot Egypt campaigned against Nubians, Libyans, Syrians, Med. Isle1, Hittites. Rameses decreed that the land Surveying and taxation. be divided among the people. t---.....-<"275 AHyria conquered Babylon and remained the great military power to 606 B.C.

1200 llycenean civilization in Crete. Its glories are found in the works of HomerJ trade with Greeoe.

1105 Chinese classic in Mathematics.

1100 Trojan War. 1056 David, King of Israel. 800 Record1 of Greek Olympic Games . DA.TES EVENTS MATHEMATICS PRODUCED ALGEBRA PRODUCED B.C. 753 Rome founded.

705- Sennachirib invaded Syria, de- 681 teated Egypt, entered Judea.

Thales--world traveler; Ionian Jlathematised natural phenomena. 1ohool. Applied deductive reasoning to geometry. Beginning of demonatrati ve geometry

605- Nebuchadnegzar II destroyed Jeru 562 1alem. "Babylonian Captivity."

594 nder Solon Greece ado ted the measurements of Chalcida for purposes ot trade.

559- Geometry, theory ot numbers. 500 Im.ported proof to mathematics. 'Discovered irrationals. Accepted and cultivated number 1 mysticism ot the East. !Began the study ot sounds vibrating I 1trings.

538 C:yru1 of Persia captured Babylon ~ byloniai:i symbol for ~.zero. Cyrus subdued Lydia and the Greeks in Asia Minor.

529- Persia conquered Egypt. 521

8~.;. {'"J. u• g1· .rttr•J.• ·.. wA.Lli J.u·~g Europe to the Danube, war with -,,.p:;jj;.._~- ~.ic ••

itially-blocked development ot •::;; ~-in Italy. ·ms.th. oy"his 4-paradoxes. Sophist revolt against plausible reasoD.ing, a cardinal turning I point. reeks .defeated Pe~siana at Greeks picked up numerology ••••• but •••• overlooked ·algebra. Marathon. Lore of Fast open rto any curious Greek. 470 ocrates ot Chios. Demonstrated power of indirect method.

429- traveler. I Center of learning shi~ed to Athen. 348 408- (second only to Archimede ) Masterly treatment of inoommensurables. By his method of exhaus tion he computed areas to any degree of pre cision required. 384- istotler his thought daninated As tutor to Alexandri a he collected the world for the next 1800 a host o:f' scientific data. . years. Unquestioned acceptance ot his doc trines retarded scientific in quiry in later Europe.

4th Bab lonian si ned numbers. Astronomical texts made cent use of the laws o:f' signs in multiplica tion. DA.Tm MlTHDU.TICS PRODUCED AIJJEBRA PRODUCED B.C. 366- Euclida professor of mathematics Deductive geometry. Geometric equivalents for 276 at Alexandria. such identitieaa · a (a + b) = a.2 + ab (a + b)~ • a2 + 2ab + b2 Sol"99d x~ + ax • a2, a pos itive, geometrically. Alexander the Great. t?f Ariatarchuaa anticipated Copernicu in discussion of solar system.

328 Alexandria replaced Athens as sen- Unequalled in enthusiasm and ter of thought. achievement until 17th century.

300 Opening of University at First organized center of essen- · Alexandria. tially secular learnings museum, library. university.

287- A.rchimedes--Syraouse in Sicily. Applied mechanics.' hydrostatics. 212 Mathematician on a par with integral calculus. almost devel Newton and Gauss. oped a.nalys is.· 3.141697 •• >pi> 3.141496

265- Apolloniusa conics. Areas of curved surfaces by method 190 of limits. 213 l!mperor of China burned all books.

150- Hipparchus ot Rhodes. Astronomy became a mathematical 100 science. 106 Pompey and Cicero born.

51 Caesar subjugates Gaul. 30 · lgypt abaorbed bJ Rome. - 4 Probable date or birth or Ch~!st.

1 Mayas of Central !merica. Possessed fully developed number system and chronology.

J..D. 64 Buddhism introduced into China.

100 Ptolemy. Fe.iled to free trigonometry--lack of algebra and disabilities of logistic.

260 Diophantus of Alexandria. First ma·de use of symbols m algebra: signs tor unknowi minus, various powers. Began to use symbols ope~ atione.lly. Diophantine analysis. Gave algebraic solutions to special linear equations in two and three unknowns, eg. x + y = 100. ALGEBRA PRODUCED DAT.EB EV EM TS MATHEMATICS PRODUCED • A.D. 200· Migration of Nations in EUrope1 SOO Vandals, Sueves, Burgundians brought on the Dark Ages. ·· . The Dark ·Ages ·were· follOvied· bi the spread of Christianity and of the Latin. language~ .. There was no science in Latin. 324 Constantinople founded. 400 Hindu matheme.tiCs became known in China.

410 Sack of Rome bY, Alari~ • __ 415 Death of Hypatia in Alexandria. European·mathemat1cs-lapsed into a state of dormancy.

4SS Attila, king of the Huns. 455 Sack of Rome by Genseric •

.. _. - 33 -

Second Period •••••••••• c. 500 A.D. to c. 1600 A.D. DEPRESSION, TRANSMISSION, TEANSITION.

Before proceeding to the chronological outline for this Period we shall outline the historical picture presen ted by the European and Moslem Worlds from the fifth to the sixteenth centuries, and how the learning of this period, such as it was, was preserved and disseminated.

Early in the fifth century Hypatia, the last to be imbued with the spirit of Greek mathematics, vainly tried to hold back the clouds of ignorance that were enveloping mathematics. When she was killed by rioting Christians in Alexandria (415 A.D.) intellectual darkness descended upon 19 Europe. The intellectual life of the Moslem world, though a sharp regression from the golden age of Greece, was much more.active than the intellectual life of Europe. The Mos- lems developed centers of culture and patronized learning, and, in general, were the agents who preserved and trans- mitted to the awakened Europe of the seventeenth century 20 the work 01· Greek, Hindu, and Arabic mathematicians. Europe showed signs of awakening already in .tbe thirteenth centur·y when returning Crusaders, the influx of Moslem learning from Spain, and the founding of Universities added new blood to her veins. (See Map 3, page 43.) But only with the coming of Vieta and Descartes did she rouse completely from her mathematical coma. - 34 -

In the fourth and fifth centuries A.D. the outer de- fenses of" the Roman Empire in the West crumbled be.fore the hordes of semi-civilized Goths, Burgundians, Franks, and Vanaals. Civilization all but vanished. The decadence of mathematics, however, began, in point o.f time, not with the barbarian invasion, but at the height of the Roman Empire. Mathematically, the Romans produced nothing. They took what little they needed for war, surveying, and engineering 21 and ignored the r·est. The barbarians overran Europe. Wandering about, they mixed with the conquered, and together set the caste o.f medieval society--the feudal system. By 500 A.D. the for ces o.f Christianity and o.f the Teutons were at work creating 22 a new civilization. Separate nationalities gradually developed, and by the sixteenth century England, France, Spam. and Portugal had become consolidated kingdoms. During the centuries of the Dark Ages Europe needed, above all, re- civilizing and reorganizing. This work consumed just about all the energy ~urope had. Contact with the Greek masters was almost wholly lost, in fact anything Greek in the early part of the Dark Ages, was often frowned upon, as Greek in .fluence was blamed for the downfall of Rome. Such pinches of Greek translations that were made were often worse than nothing at all, being translations by scholars who failed to catch the spirit of deductive mathematics. Translations

we~ often watered down to serve as textbooks .for the teaching of the rudiments of arithmetic and counting. The men - 35 -

who kept mathematics just breathing in Europe were not mathematicians at all; they may have been scholars, theo- logians, logicians, teachers; but they were not mathematicians. From the founding of Monte Cassino in 529 A.D. until c. 1200 A.D., the Christian monks carried the main burden of keeping and spreading the flickering light of learning in Europe. The Roman Church meant to Christianize and eau- cate the barbarian invaders, and sent missionary bands among them. These bands founded monasteries, often deep in hos tile country, which became the nuclei of' religious and educational work in that district. A very important work of the monks was the copying of manuscripts, a long and tedious task. Later, as the countryside was Christianized, village and city churches were built, to the larger of which were attached cathedral schools. Many of these cathedral schools grew to university proportions in the thirteenth and fourteenth centuries, and then took over from the monasteries the control of copying and selling books. In the early half of the fifteenth century the manuscript trade became more general and separated from the schools. Printing from movable type in 1450 turned out books by the hundreds and thou sands, and the sack of Mainz in 1462 benefited Europe by 24 scattering competent printers to other cities. Printing from movable type was the second major advance in the dissemination of knowledge. The first great step in spreading learning was taken when the spirit of free thought in Greece struck at the traditions that kept knowledge from - 36 -

the masses and made it a prerogative of a select group. The third step came in 1826 with ·the editing of technical ·25 periodicals. In 632 Mohammed died, he who had started ·a religious revival, the like of which the world had never seen. · With fire and sword his followers pu~hed the Eastern Empire back to Asia Minor. In short order they conque~ed Persia, Ara bia, India, Egypt, North Africa, and Spain. As they were about to sweep Christendom from the map, Charles Martel cut short their conquests in the West at Tours in 732. In 1453 the Ottoman Turks~ after being held off for eight centuries, fought their way into Constantinople, and took over. the By za~tine Empire. They continued into. Europe until, in 1683, Sobieski of Poland put a halt to their march at Vienna.

Mathematics fared a little bette~ during these cen turies with the.Moslems than it did with the Christians. The Moslems did not have barbarians to assimilate, and, /though their.enduring influence on mathematics is questioned, they did encourage learning in the countries they conquered• Bagdad on the Tigris (750-1258) was a center of ·culture in the East, to which the: Caliphs brought scholars from dis tant countries. Cordova in Spain became the center in the West. The Moslems 1 great work lay in translating Indian . and '·Greek mathematics into Arabic and Persian and in their attention to algebra. By 1100 Europe was strong enough to send Crusades a gainst the Moslems. The Crusades failed to wrest for long - 37 -

the Holy Land from the Moslems, but they stimulated trade and the growth of towns from which developed a class of merchants and freemen, who in time were to overthrow the feudal system. The Crusaders, moreover, carried home with them a fair supply of mathematics in addition to the other items of interest they picked up. The Crusades did for the

~urope of the Middle Ages what Worlu Wars I and II did for the America of today: thousands and millions learned how people in other lands lived. (For the routes of the Cru sades see Map 3, page 43.) In the century preceding the expulsion of the Moors from Spain (1212) scholars from Europe had visited Spain, openly or in disguise, in search of the learning they knew was flourishing there. The twelfth century thus became a century of translations for Europeans, translations from Arabic into Latin; just as the ninth century was a century of translations for the Moalems, translations from Greek and Hindu into Arabic. The thirteenth and fourteenth cen turies saw the rise of Universities, and the influx to these Universities of Jewish scholars from Spain, scholars well versed in Moslem learning. (See map 3, page 43.) The fif teenth century saw the advent of printing and the voyage of exploration to America and India. When Constantinople fell to the Turks in 1453, the Greeks in that city fled to Italy and brought with them the classics of literature the

Arabs had passed by in favor of ~he Greek scientific works. - 38 -

Counter-balancing these spurts of learning and almost can- celling them completely were the Hundred Years War between the two most advanced countries in the North of Europe, the Black Death took toll of from one-third to one-half the po- pulace, and the Mongol scourge from the East that ravaged 26 the eastern half of the continent. During the centuris of the Middle Age a the Chine ae and Hindus were somewhat active in mathematics, and from exis- ting records we see a transfer of knowledge from West to East and from East to West. The exchange of ideas in those days was made through the tradesmen, pilgrims, armies, and 27 embassies. ( See maps 3, and 4, pages 43" 44 • ) Glancing over the eleven centuris from 500 A.D. to 1600 A.D. we are appalled at the little progress mathematics made. The only advances of real significance were the gra dual acceptance of the Hindu numer·als and o:r the zero. Ex- amples of Hindu numerals, in forms resembling our own, were found in India on inscriptions dating from as early as the third century B.C. They must have been used rather common ly because the symbols for the num.tlVals underwent successive 28 changes in form. The :first extant external reference made to the Hindu numerals was in 650 A.D. by a bishop in 29 Mesopotamia. A Sicilian coin dated "1134 Anno Domini", is the earliest known application of Hindu numerals in 30 Christendom. The Arabs became acquainted with these numerals in the ninth century, and Fibonacci advocated their use in Europe in 1202. The zero came into its own in India - 39 -

in 876 A.D., although it was used much earlier than that. The Babylonians had both the idea of zero and a symbol for it, and the Mayas in Central America used the zero, possibly 31 as early as l B.C.

In the chronological outline for this period under- the heading "Mathematics Produced", we note references to tri gonometry. Today we associate trigonometry with surveying and electricity, but those who founded trigonometry developed it as a tool for astronomical work. The Moslems were very much interested in perfecting their astronomical instruments and their astronomical tables. They did apply a little algebra to trigonometry, not for the sake of analysis, but 32 for better or quicker computation of tables. Because the word "algebra" is of Arabic derivation we may be led to believe that the Arabs and their teachers, the-Hindus, were excellent algebraists. They may have been for their time, but they were not algebraists in the modern sense of the term. Authorities differ in their evaluations 33 of the algebra produced in this period. An elementary text of algebra today contallis little Hindu or Arabic al gebra, in the form in which they used it. While algebraists of this period spent their time investigating quadratic, cubic, quartic, and simultaneous equations, their solutions were special solutions, arrived at ingeniously perhaps, and by hard labor, but their methods are of little worth today, when weea.sily handle whole sets of equations by general me- - 40 -

thods. The ground-breakers did not even envisage general solutions. In 1770-71 La.grange set for himself the task of' investigating these special solutions with a ~w to giv- 34 ing them a unified treatment. He succeeded. Early algebra was rhetorical (written out in full), and only after passing through tre stages of using abbre viations for words (syncopated algebra), did it come to symbolism pure and simple (sj:m.bolic algebra). The Hindus i:hrew a broad hint to the Arabs ,--the hint to develop oper ational symbolism. And what did the Arabs do? They not only did not develop operational symbolism, but also lar gely rejected the use of abbreviations and went back to rhetorical algebra. In addition to being hampered by poor symbolism, the Hindus and Arabs were plagued by the lack of an extended number system. They were forever rejecting negative and imaginary roots. The Hindus, however, developed manipulative skills far superior to those of Diophantus, the Greek algebraist, and they made an honest attempt at 35 ·symbolism. The Moslems built for themselves a reputation as translators, commentat6rs, and minor contributors. Their scholarly translations were perhaps the spark the seventeenth century needed to set itself off. It would be interesting to speculate as to whether or not mathematics would have developed as it did had the mathematicians of the last few centuries started from scratch, instead of from the works the Moslems had accumulated for them. - 41 -

We mentioned that this period was one of transmission, not of production. From whence did the important ideas first stem? Did China influence India, or did India influ ence China? We know the Moslems translated Greek and Hindu works. In their translations they acknowledged ~eir in debtedness. But the question of priority among the Chinese, Hindus, and even the Babylonians, we are not able to answer conclusively. In India, for example, the records before 36 the Moslem invasion are not to be trusted. There had been developments before the seventh century A.D., of that we are sure. But to greak down the records and say this man developed this at a given date, we need more accurate data than existent records give. And China 'l All books were or dered burned in 213 B.C.1 However, after 450 A.D. there are references in the Chinese annals to peoples from t'oreign countries. Some of these references are incorporated in 37 the outline.

• • •

In ancient times the two main channels for the transmission of knowledge were wars and trade. To these, in the Middle Ages, were added pilgrimages, the exchange of em bassies between countries, the rise of the Universities, and printing from movable type. The most important wars from the standpoint of extending a culture were the Moslem conquests and the Crusades; - 42 -

.. and on Map 3 are indicated the extent o:f the Moslem con quests and the routes of the important Crusades. After the Moors were driven from Spain, the learning they had fostered spread over Europe mainly through the universities. On Map 3 are located a number of the im portant Moorish and Christian universities. On Map 4 are traced the trade routes of the Middle Ages, both overland caravan routes and sea routes, which linked the various countries together. On Map 4, like wise, are located the teaching centers of some of the out standing Indian mathematicians. - 43 -

MAP .3

EXTENT OF MOSLEM CONQUEST . EXTENT OF EAST ROMAN EMPIRE . 38 MOORISH UNIVERSITIES: Cordova Seville Toledo CHRISTIAN UNIVERSITIES : Oxford Paris Cologne Leipzig Prague Padua Bologna Pisa CRUSADES 39 ++++++ + FIRST: Godfrey de Buuillon Raymond of Toulouse North French Normans ------SECOND: France and Germany •••••••••••• THIRD: Frederick Barbarossa Philip Augustus of France Richard of England -4-3R- - 44 -

MAP 4

CARA VAN AND TRADE . ROUTh S OF THE MIDDLE AGES 40

Water routes . ++++ ... ++ Routes developed by the Mongol conquest . Pre- Medieval Routes . Arabic route from Persia ••••••••••••• to India . Land routes developed in Middle Ages in Asia and Africa.

TEACHING CENTERS OF HINDU MA TilliMATICIANS 41 Patna ••••• Aryabhata. Ujjain • • •• Brahmagupta. Mysore •••• Mahavira. Jaipur Benares Bombay Poona Madras -4411 - 45 -

CHRONOLOGICAL OUTLINE OF THE SECOND PERIOD

c. 500 A.D. •••••••••••••••• c. 1600 A.D.

It may be well, at this point, to reread page 26 before proceeding with a study of' the chronological outline which follows.

' j "f i " ! t' DATE EVENTS MA.THEMATICS PRODUCED ALGEBRA. PRODUCED ··A.D. 475.;. Boethius (Rome}. Boethitis- compiled what. -tlie Roma s 524 h.ad .. trarisini tted · ·of. Greek: ·ma.th His arithmetic was taughtiri"' ' the'' schools during' the' Middle 529 St. Benedict founded monastery Agffa. ·Drew hiS-· arithmetic at Monte Casino. 'from Nichomachus~--wh.o drew his from Pythagoras. 554 Korean scholars Introduced . Chinese mathematics in Japan.

6th Aryabhatta (Hindu). Simirii.ed arithinetic progre'ssfons •. cent solved determ:Lne:t:;e ·quadratics· ·in· one unkiiom1; ·s·o1ved ·· iriaete:tiniri.:. ate 'linear equations in two un 'kiio\ms. - .,.. .c • Sugge's"ted use 0£ l'etters to represent unknowns. 615 Arab embassy vi_si~ed China.

622 Death of Mohammed; Moslems conquered Persia, Iridia,, Egypt, ?forth Africa, Spain. ' - 636 Roman priest in capitol of China.

645 Huall..,;tsang brought 657 Hindu works back to China.

650 Hindu numerals. First external reference to I:Iin u numerals was made by a bishop (

672- Bede i;he Venerable \¥n£>.). C~u~~::t~i~•i•; ~;..,; I 735 ·· · .ringer· rec1'oilinf;:~ ~one· monk in

,, __' -~------each inonasteri 11/as to· det=e=r'-._--lf>-'-----,------,----__,.---- __ niine days' of .religiou"s f'es.;. tiviils' 'a:iid compute the 'calendar.·

7th Brahmagupta (Hindu). Algebraic'ruliia · for"nega:tives ~ ·a'6- cent. taiiieC1 one root' £or' quadra tfo's. - Gave complete· 'integer·. solution. c£ a.x ·i:· 'by =.. c ; a:, b~o ·constant iii -hegers~-used abbreviations f'or iii:iknowns ; · · ·~ · .. ·· · · · .. - . Hiridii"algebra at its best was. stfll ' largely' rhetorfoiil"in that aper~ tional directloris"were· ri.ot- fully symbolizea~ Lacked extended num ber system. 712- Foreign ships visited China. 825 719 .Ambassador from Rome in China.

732 Charles Martel halted Moslem advance into Europe at Tours.

735.;. Alcuin (Ire.) directed ·educatiolll In the great sees and'moruister 804 in the Frankish Empire. ies he fcitindedschools-in which were· taught psalms-, · - writing,· singing, ·computatio arid grarmnar.· ·nra:gged numbers i:i:ito' 'tlieo1og·y, 'as did many other scholars. DATES .. EVENTS MATHEMATICS PRODUCED ALGEBRA. PRODUCED A.D •

. 750- Bagdad on Tigris-.:.center of Astroriomy· was -develc)ped. tO take ca.re 1258 culture in East. . . of 'religious 0 hservance; - .. 'k" ••• ·- • 1) Prayer-a ·said. £aCing Mecca. 756. Cordova in Spain--center of 2) Prayers and ablutions at definite culture""iri Vlest., especial ·· hours. -- · -·· · ·· -·· .. _ .. - ly after 1000 A~D. 3) Eiac·t'dates of feasts. 4_LJ\~~!°.~~g;r_· _____ .. --· Ce770 Hindu astronomer summoned to TrariSlated cifrtain.. trig tables· into . Bagdad. ;A.:rabfo'. 'Hiiid.u numbers entered Arabi iriEi.thematics.---·Arabs perfected.. astro rioinioal ta.bras· and· ·1n.strlimeiits., and e'rected'' observatories·;· Ma-thematicia was first of all astronomer. ~-~··~······ --·~-·~----.- 813-· Al-Khowarizmi (Arab). Wrote. a book- ori-al'gebra -a:n.a··arithme Gave· algebra Tts ·name: .. c.850 tic~ · · Trari.518.tor~-- commentator., a.l;.;'jebr w' almiiquabala.:..:. minor contributor. 11 restoratic)n and' ·reduction' His algebra was rhetorical. 814 Empire of Charlemagne fell apart.

9th Hindu numerals became kn.own cent. in Arabia.

. 876 Zero • First undoubted occurrence in India.

9th Mahavira (India).. Discarded-ima.giriaries· 1n. his.. cent. solutiOns~ Hindus -lacked an .extended number system. 915 Mas' °Udi, Arab geographer ·and historian., visited India·". ·~·· ·~ · · . Ceylon., and China. · ····-·"·--~-·9··2--9~--'1-A~l--__· B_-a_t_t_·~-- __(_M_· ~-~--~-~~-· -_)_-~ -· ___:~ T~~ia,~i~!~i~~~d~-~~:it1~~r~i~~=. nometry. "Iii-addftii'on yto-lliiidii. sine., he· used tangent" arid'. co"tiangent'~. - -- . - 10th In 'trigOnometry~· 'iie·c-arit. and-cosecant .. .. ·- cent. made their-appearance~·- ·conception · ra.· func"tl.ori .. still soo years in the· ftiture .- The trigonometry of these times was not much like the trigonometry of today. ·

1000 Al-Karhi of Bagdad. Greatest (?) algebraic work o.f" the Arabs·. His algebra was rhetorical.

1003? Gerbert (Pope Sylvester II). In hls time the -ocCideiit ca.irie T1ito sure possession· of all the i:riathe :ina tical 1m6wledge ··or the Romans.,· ~hic_h _was Vf!'r_Y. li tt~e.

1050? Abul-Wefa (Moslem). System- of· trigonometry into a loose deductive system. 1066 Norman conquest. 1095 ·First Crusade.

1114- Gerard of Cremona. Spent ·nearly· 50 ·years·:in .. Spain·. trans::. · 1187 Ia ting ·meaicai.- astr.ciiiomfca.1, phil osophical; and mathematical treatises. DATES EVENTS. MA.THEMATICS PRODUCED ALGEBRA PRODUCED A.D. 1120 Adela.rd ot Bath (Eng.). Traveled to.Syria and brougllt back ''a 'resume of'" f'oreigri 'mathematics. Tra:riS1ated Eiic11a··mto La.till; - · Trfuislated· Al""Khawarizmi' s astro- -nomicar tables. - · Translated 90 Arabic texts.

c.1123 Death otomar Kp:a~. Corrected the-calendar oy the-in.;. Wrote a t'reatise on algebra in which Persian poet-mathe trodu?tion of' cycles of' 33 years his geometric treatment of' the matician. 'cubic is''ceiitral; . Refused riega tive. roots• All positive . 'roots: not ·:round.' ., .. .. ' ' ' . ' .... ' . Failed'tci'come to grips with the num ber concept.

1100.;. Crusades. Europeans i;irofi tt:id much -by-- tneir 1300 contact wi th'~Araoian culture~---. - then·-fa·r superfor tc) ·their oW?l~ Perhaps tne· greatest 'result was th establishment· of u'niversities: ·· - P8.ris~· Col'ogne; Prague, Leipsig, ··· Bologna, ·Padua,· Pfsa; ·· O:iford-. · · 1146 Second Crusade. The 'study o:r matheriiatiOs was main tained b.alf-lieartedly." No. great teacher-matheinatiCiari·appeared to inspire the students.

.12th Bhaskara (Hindu). Recognized two roots for quadratics, cent. - but discarded negatiws. . · Gave m~thod for finding:particular· . solutions .. ot·x2 --Ay2':· 1,·A•8~6l, 67,92, and found general solution.

1201.;. Nasir-l!iddin '-:~~~.:__~1~7~-"-· ·-~~- .. Le~nardo,-de ?!ila/> · · · -tiber:Abac~·-~~-cbn.ta:1n.s:-~11··:~f~6-·movii·; · F.i.bonacci.(Italian) edge the Arabs possessed iii arith;.;. · · As a mercha.llt he· metic 'and algebra~ but nO advances. traveled· extensively Advocated-Hindu ·notation·: abacus and in Syria, Egypt, ·· · coi.mtiiig' board·' discardea.··- ' Greece, and Sicily. Liber Abaci was for centuries the storehouse from which authors got matefrials" '.for works on ari tbmetic ·a.n.a ·a.1gebra. -- ·· · · · - 1204 Sack of' Constantfiiople Fibonacci attempted to prove that a by the Crusaders. · geomeifr:i.cal 'coristruotio:ii of a_ root by straightedge arid.compass aione 1220 Chirighiz·Khan's great l'o?"·x3 +· 2x2 + lOx = 20 is· im expedition to Europe. possible• Nothing. like this Mongol scourge .. in .. attempted proof' 0£-:il'.!lpossibili ty Asia and nearly half until the "19th century~ . · of' Europe. · He failed.to sense the generality of a problem.

1220 Leonardo published his Work. on indeterini:i:ui te analysis. Y.ade Liber Quadratorum •. him the-outstal:idirig mathematician .from Diophantus to Fermat.

1225 Leonardo published his · Cori:b.iiris· all the· knowledge. of - · Practica Geometriae. geometry· and ·trigonometry trans mitted to him. 1256 Caliphate ():f' Bagdadcon quered-by_Mongolia.n hordes. DATES \EVENTS MATHEMATICS PRODUCED AIGEBRA PRODUCED A.D. 13 cent. iranciscan and Dominican Orders 13 cent. The "reckciri.i:ilg on lines" began' and extended all over Europe. 1265-1321 ia11 :Marco Po~o_begins his travels. 1300-1350 Invention of Gunpowder. GuilpO-i.•ider riecessi tated much re firie'd mathematics ail.cf higher. dYn0.miCs in tlie-aocur&.:te.cal culation of trajectories.

1304-1374 Petrarch (Italy).

1338-1453? Hundred Years War (France and England). 1347-1348 Black Death: 1/3 to· 1/2; the· population of Europe died. 1390 Uniform-weights and measures in IJ10St_ o_f' ~gland. __ ...... __

1400 During the 1_4th and 15th. centur~es matherii.a.tical sC:i.erices were· .. ·. . almost at a standstill. Gerinii Emperors ·and· Popes were quarrel ing: Guelphs vs. Ghibellines ~- · France and England. were at war. England had the War of the ~o es. 1420 Era of the Medici in Florence • . ·-- ......

.1436-1476 . John Mueller, Re&iomontanus First Euro&ean ez~ositioll ot . . \· (Germany). 1 pl~~ a~d: _s_ph~r~cal _~_rig~~om-l i ~--~-:-o------·--···-·- ~--·-~-·· -- ______,_____ L. ....~t_:ry_~-,,-~~Y,,.~se~ si~e and cc;:_·t····· ______.. -----·-··-··- sine. Did much to separate l trigonometry from astronomy • .... --~- - __ ..,_ --~~-.. -· - ~"-";'.~· 4 ' , ... •.J"- _.,. • ....., ...... L _,,_,._·.,.•.-'""•~A~ ._.- ... V.~···". I.

1450 Invention of Printin~. Book·s· for-in.any-." rn· first· 50 "yrs Printing 'insisted on ·uni;,;, Italy produced about 200 book form simplified symbolism. on mathematics; during· the next c_e11tury about 15QO •.. 1450 ·Robert Recorde (England). . The ·arovnd- 0£ Artes: orie of the most celebrated of English textbooks.

1452-1519 Leonardo de Vinci (Italy). Painter, s·culptor~ ge.rieral scie - tist,-architect, writer, all aroundgenius. . .. . -'.. ~' ~ .. , . 1453 Constantinople captured by the .Learned Greeks ffeu to -the West Turks. and directed translations.

1462 Sack.of Mainz by.Adolf of Nassau scattered printers over all of Europe.

1473-1543 Copernicus (Poland). First to break.away from the 11 "Aristotle-worship • · · · Copernican theory of the solar system. 1475-1564 Michelangelo (Italy).

1483-1520 Raphael (Italy). 1483-1546 Martin Luther (Germany). DAT.ES EVENTS MATHEMAT_ICS PRODUCED ALGEBRA PRODUCED A.D. 1486-1567 Stifel (Gennan). · Did £or Germany what Card.an and Greatest· Geririiill algebraist of. the Tar.~~gl_~a ~id f?:t". -~~al~~- 16th centiicy-. Gave· 'tlie germs of the theory of exponents. c.1489-1559 Adam Riese. Greatest of ..the Ge:ririful Recheiiineiste s of the· 16tli century~. Replaced . coiri.puta'tiori by counters by writ 1491-1556 Loyola of Spain. ten computation.

1492 Discovery of .America. Implied possioilit:i.es- for mathema tics: accurate :riavigation-;;.113..:.. place's ·celestial" mechariics of th . 1497 Cabot discovered-coast 19th c,entury;· Eula·r viorkea for th of North .America. British Admiralty making reliable tables. ????-1526 Scipio Ferro (Italy). Solved :x3 ·+ mi = n in.1515, im parted result'fo pupil; Antonio ..:£il.o_r_ v1_i_thout publicatio~. Discovifr1es· fo.-tllese dayri· aria· for 1501-1576 Cardan (Italy). tW6' centuries 'after 'vere'··kept Ars·UB.giia:-suriiall.d.crowil.of all secret.to secure an advantage algebra uj:i" to.his.time. - , over rivals. He~ calls negative roots fi'cti fious; i>ositi ve- "roots' re~l; imaginary ··roots;· impossible. ·- ·· · · -- ··· · First solution of bi-quadratics • • ' • ,,,_ < - • •• • ' "' • "' -~ 0 ... • • Mo. 1506-1557 Nicolo of Bresia, Firs't to. apply'mathematics to mil Solved . x3 +· p:x2 = n', in addition Tartaglia. ":i. tary' scienc'e. "' . -- . ' . - ····· . ... to solVing · Ferro' s form." Cori.;.. Treatise"· on arithmetic: best of his fided ·his' solution to. cardan~- 1509-1564 Calvin (Switzerland). century. who published it as his own in Ars Magna. 1519-1522 Magellan-circumnavigateld the globe. _ . ~·' - • •· ·•-•• ~· •~• •' •• • • • •- •·e . .,. ·-·.,~·• ·""..,. 1540-1603 !.Franciscus Vieta (Fr. »·I , IThe tra~si t"ion from· th~ ~peCial to ------·-·· ··--·------··------····-·•-·· - ··---··---·----·----- ______,______.. __tp,~::_~~rr~r~l~:i!'.i-ffrflt .. ~i~6~!"!!1hl~ in his wofok.Iii his- solution of l quaaratfo,- ·cuoic; · s:nd.· qtiartlc:-· equatiOns .. he ·1-e·auced ·cubic ·a.n.

1548-1620 Stevin (Belgi~~· Investigated statie1? aria· hydro.:. - · statfos.- Advocated· legal adoption 1550 Council ii£ Trent:· Ro~/ . of decimals in 1585. / man Church reform·. -'. 1550-1619 Napier (Scotland). Logaritliiris,-Napiers Rods~ Rule of circular·-parts in spherical triangles.

1553 University of Mexico.

1556 Juan Diez (Mexico). summarfo coin.pendoso, - first mathe matical text of the New World.

1558 Elizabeth, Queen of I England. - 51 -

Third Period• •• , •••••• c. 1600 A.D. to 1700 A.D.

BEGINNINGS OF MOr.ERN MATlillMAIJ.1ICS. / Bei'ore ·passing to .the .chronological outline i'or the third Period, we shall give a brief' picture of' those phases o:f the universal history of the_ seventeenth ceritury .that - . ··- - were related to the development ·of _mathematics.

· The· spirit of ·the.:age during .the seventeenth century may be su-rized ·in three -isms: commercialism, national . 42 ism, and individualism. ··Each of these is related to the development of mathematics. ·In the sixteenth· century great attention was given to the· arithmetic· o.f commerce. The early teachers of commer cial arithmetic were often textbook writers as well as teachers. ·These teachers, however, were not necessarily creative mathematicians. Men like Robert Recorde in Englani and.Adam Riese in Germany were teachers, not mathematicians. With the discovery of America and of a water route to India· the center .of commerce shif'ted from the Mediterranean to the Atlantic coast. ·spain_and Portugal.had struck gold.

. . ' early in the·Americas. Portugal built up a lucrative tratle vii th the·· East· Indies, and these two countries :for a hundred yea-rs dominated the colonial world: Later the Dutch con quered· the Portuguese East Indian colonies; and Spain lost

. ' her. control 'of the -seas with the sinking of the Spanish Ar- mada in·i55B. The· Dutch·~ French, and ·English now entered the' colonial field, and the center :o:f ~ommerce 'shifted"

I. 52 -

farther to the north. Italy had faded into the bacl{ground with ·the opening of the trade routes. in the West, and France and Spain spent a large part of the :sixteenth cen tury fighting over a country.whose riches had waned •. As commerce increased Western Europe grew rich, and life in creased in comfort for ever growing numbers, conditions ·43 favorable for the development of art and science. As Spain and Italy lost their positions as key powers in the world domination Of politics and trade, their productivity in the: fields of art and learning practically ceased. The great men in mathematics in the future were to come· from the nations in the north of Europe, France, England, Hol land, Germany, nations that were rising, politically and commercially. · No :age· in history has been· free from wars, and the seventeenth century was no exception with the Thirty Years'

War in Gerlna.ny, the civil war in England {1644-45), and the War of the Spani,sh ·Succession at the turn of the cen tury. But these wars_ were local affairs, not wide confla- · grations like the barbarian invasions, the Moslem ~onquests, the Mongol scourge~ During local wars learning in the areas of conflict drew itself apart to the shelter of a few men, more interested in the pursuit of kriowledge than in the kil- ling of fellow men. Once the fighting had ceased, the in stitutions of learning were rebuilt. As all the countries were not at war together, all the centers of learning were never eliminated at one and the same time. The barbarian - 53 -

invasions, on the other hand, attacked all fronts simul- . taneously, and the time required to recover was proper- tionately longer. As a result of the Thirty Years' War the Dutch and the Swiss,· in 1648, gained their.: indepen- ' .. dence from the Roman Empire. The German princes were left almost independent of the Emperor, and the Protestant elec .tors of Brandenburg began to lay the foundations of a mi- .. litaristic Prussia. Meanwhile, in· the east of Europe, Peter the Great was building Russia. into a powerful nation. Though the spirit of nationalism may be. looked upon as a form of individualism, individualism- of a group, the spirit went much deeper. Once the masses accepted the idea of questioning the authority of the Church, the natural .thing to .do was to question the authority of the rulers and \ .the authority of learning. Luther, _Calvin, and Henry VIII of England had led large groups away from the authority of the Church. These sects broke into other groups as indivi . duals questioned the authority of Luther, Calvin, and the Cb,urch of England. The. American colonies were, to· some ex tent, a.means of escaping religious persecution in Europe. We find the questioning of secular authority reflected in the establishment of the English Parliament, which .became .dominant in English. government in 1689. France had an

Estates-General, but it was suppressed in 1614 to be sum moned again only in 1789,. when the relations between the

royalty.and~the masses were so strained that it was impossible to avert the Revolution. The .power in the new - 54 -

Dutch Republic was invested in a States-General. In the seventeenth century already the trend in·governmerit was. ·44 towards more and more rule by the people. The spirit of individualism, or the questioning of autho_;i:ity, did not confine itself to religion and politics. Aristotle, Ptolemy, Euclid, and other Greek masters-had for \ . centuries been the unquestioned authorities in.learning. Their conclusions regarding natural phenomena were far from

b~in supported by the many new facts now uncoyered •. Scien

tists were inventing new instruments for inve~tigating na ture a the .teles,cope, the microscope, the pendulum clock.

They_ were 0making many careful observations and drawing_ con-. clusi,ons, only a:fter experimentation. The blind acceptance

of Gre~k authority in science. had retarded the advance of science for centuries. Science was now .free and.it advanced rapidly. During the seventeenth century tb,ere started a movement that developed moFe _fully in the eighteenth; . the calling of scholars from country to country by universities, aca- ~" . . " ' . . ' ~ demies, and royal command. Eminent scholars thus became the product of not· one country but many, and this exchange did much ,to spread and.stimulate the growth of learning and ,45 culture. . De.scartes, in 1647, accepted the. invitation of Q.ueen Christina of. Sweden to join her Court in the role of . . 46 p?ilosophic tutor. Galileo accepted ~he office of Mathe- 47 matician and Philosopher to th~ Grand Duke of Tuscany. • ' ' ,, • • > ~ 48 Leibniz was librarian to the duke of Hanover. - 55 -

The relative ease with which books were being printed makes it difficult to trace the influence bearing on mathematical developments. Authors had large audiences in almost -every country, and the inspiration found in one book led.to ·the writing of· others.· Tlie mathematical front was widening, and c6ntributions ··were being made from various parts of

·Europe. An important refinement, demanded by the printing

process itself, was the ~doption of a more uniform and prac tical ·symbolism than had existed before printing, when each· individual was free to express his originality.

-· - ~ ·Modern mathematics, as may be seen from the. chr9no-

logical outline, 1 had its beginnings in the seventeenth cen- . tury. E. T. Bell calls the h~lf-century between 1637 and 49 1687 the "fountain head-of modern mathematics". In 1637

.Descartes published his Geometrie, and in 1687 Newton his

Principi~. The new and powerful metho.ds_ of Descartes 1 ana- 1lytic geometry and Newton's· calculus at one stroke raised ··mathematical endeavors to a new level. For over a hundred years the best mathematical minds were kept busy with the .wealth of mathematics these-methods untSJ"overed. The methods themselves were clarified)and expanded, on. the one hand, and .·applied to the field of physics on the other.. Physicists, once· limited to the study of statics, now turned their attention to dynamics--an immense unexplored world. The stu- .dy of dynamics and the machine age which developed from it had important social implications. - 56 -

The five major advances of this century alone outweigh witli ease '.the sum .. total of the mathematics developed during the. previous fifty centuries: · the analytic geometry of Fe.:rmat .and·_Descartes; .the ·differential and i:tegral ·calculus of Nevrton and Leibniz; the :.combinatorial analysis and the theoryof probability of ·Fermat and Pascal; the higher arith- .. - . metic of Fermat; the dynamics of' Galileo and Newton, and the latter's principle· of universal gravitation. Two other im portant.lines of thought of the· seventeenth century, that were.· shelved until the nineteenth and twentieth centuries for development, were the synthetic projective geometry of De- . . ' sargues and Pascal and the ·beginnings of symbolic logic by Leibnitz • . .Mathematics had come. into .its ovm again. Somewhere,, somehow during the centuries of dormancy it had changed. The mathematics of Archimedes, Euclid, and Apollonius was syn thetic; that of Descartes, Newton,, Fermat, Pascal, and Leib- 50 niz was analytical. We may ask ourselves.at this point whether or not ana lytic geometry and the calculus were the end products of a process of growth that had been going on since the ancient Babylonians first started to count. Bell answers this ques tion in the negative. He consulted men who are expanding mathematics today and is of the opinion that the gradual 51 growth explanation is inadequate. Descartes did ~uch more: than gather all previous ideas in algebra and geometry and ·~ ., arrange them in a way to produce analytic geometry. His was -· 57 -

a st.roke of genius that fused two separate fields into one which contained both, but had a life and vitality of its ovm. The history of mathematics is a history of men bury ing themselves in blind alleys and details,. with an occasi

o~al genius who sees a generaliz~tion that sweeps up the works of a hundred others. Lancelot Hogben stresses mathe- · matical development as resulting from the social pattern of a given age. The Babylonians and Egyptians went so far and no farther because knowledge was the exclusive prero gative of a select class. The Greeks were held back from the development of the calculus by the social inertia of their· clllinsy number system. The seventeenth century was an age ·.of reaction 'and questioning of authority; and dissatis- 52 -faction with the status quo stimulates advancement. - 58 -

CHRONOLOGICAL OUTLINE OF THE THIRD PERIOD

c. 1600 A.D •••••••• ·'·• ••••••• 1700 A.D.

It may be wel:J.., at this point, to reread page 26 before proceeding with a ~tudy of' the chronological outline which follows. , . . '

DATES Ji.VENTS MA.THEMATICS PRODUCED ALGEBRA PRODUCED · A.D. 1558 Augsburg Confession: Protestantism recog nized in Germany. ''·, 1559 Buteo. Solved siin.ul ta.neciu-s aqua tioris -by addition and subtraction as we do' riow~ ' ' 1559 Thirty~nine Articles:· Intrciduced the -use 'cif--dii'ferent letters Aiiglfcanism establishe for different unknowns. in England.

1564-1642 Galileo (~tal;y). Founde·r -- of the sCience -of d.yriamics. Hi6":V16rk later re 1565 Fouriding -o·f St. Augus.:. ·sul ted -in Newton's Law of tine,Florida (Spain). motion • .. . '. , ... 1570-1630 Kepler (Germany). Introduced a new.idea Irito .geometry: the irif'iriitely great.and'the-infiriltely small~ Laws of Planetary Motion. 1585 Harriot (England). One'ci:f' the-founders of algebra, as we Surveying .. and mapping know 1 t today. · in N. Carolina. His work-helped to set the standard for textbooks. 1588 Spanish Armada-: England becomes greatest sea power.

1593-1662 Desargues (French). s~thet~c proje_c~~ve .geometryr ... __ _ 1596-1650 Descartes (French). Analytic geometry. Improved algebra as an abstract science ·by the systematic use of exponents and ~ a. full interpretation and construc,tion . . t of negative quantities. ______:~Excell_E1E-j:; 1 _sYJilbQ.lism: o\!t_present indices i for powers;' use---of 'x,y. z- as''varfables;___ _ use of a,b,o as constants. 1598-1647 Cavalieri (~~al~). 1601-1665 Fermat (French). Theory of 'prob-abiH ty. Greatest-niatliematician of th

I 17th century. 1607 Founding of Jamestown. 1608 Invention of telescope • '>< 1614 Napier published his l?garitbms. 1619 Speidell (German). Published logs. to base e. c.1622 Oughtred invented sliderule. 1623-1662 Pascal (French). Theory ar· ·prohability. P~?jecti!~ geome~~Y· 1624 Briggs ·published logs. Usable .. log tables;-- By ·1630 to base 10. they v1ere--stari.aard--equip ment for astronomers. 1627-1691 ' 1629-1695 Huygens (Holland). Pliys-fois·t~- ·astronomer, mathematician. c.1630- Fermat. Higher arithmetic. 1665 DATES EVENTS MA.THEMATICS PRODUCED ALGEBRA PRODUCED · A.D. 1631 Oughtred (Eng.) published Work on arithin.etic 8.nd'algebra. Clavis Mathm.aticae. &phasized use of s;Ymbols, three of. which are still in use: x, : : ,"'-'. ~ " . - - -- - , . ., ' . ,_. - . .. - .- -· ... 1631 Harriot (Eng.) ·published· Foriiiation of equations with given Artis Analyticae Praxis. roots • ...... Statement of relation between roots . arid coefffo'ierits ~. Symbols:')~ < • 1636 niversity of Harvard. Solveid ·c.i.uadrati'cs ·by· facforii:ig, ··ig noring factors that gave negative 1637 Descarte·s published his roots. Geometrie.

1642 First· adding- mifohine in vented by Pascal...... ~ 1642-1729 ewton (England). Diff'erential"i.ind integral ca . f'orins of the· calculus and their applications; £or the next 100.. years, dre\v tne attenti6n of' 'the" bestzriirids to the coiriparat"ve neglect of' algebra, except for im- provement.in ri'otation. · -- .. · · · ··· · · · - .... Universal gray~~at~o!l_• ... Discovered binomial theorem.

1646-1716 eibniz (Germany). Diff'erential -and· integrar . ·an.eralized · NeWton' s "binomial theorem. c'alculus: long' sought .for. Important contributions to notation. method' of 'irivestigatirig con tinuity.inall its manifes tatfons~ · · · · · · · ··· · · · · · Beginriing of'-sylliooHc-1ogic. Combinatorial analysis.

1654 rm.at and Pascal. Combinatorial analysis:~ theory of probability• ,Only r 50 Years after its creation i I the theory of probability I ------•---'------L__wa.a_in_Wie_on_mQ:r:-.t.it..H:l;_y ~ 1 tables. Usea··-t~day-in _n...,.u_c,,__=i=::=::::::..::::::::::-:-::~~ I lear physic 8.. __

1654-1705 Bernoulli (Switz.). Developed' the calculus,· ana lytical geometry, probabil ity. 1656 clock invented by

1661-1704

1662 oyal Society of Londo~~-

1666 ench A?ademy of Science.

1667-1748 ohn Bernoulli (Switz.). Phys'icist,-chemist~ astrono.:. · mer. ·Helped spread the cal- . · culus in Europe. 1673 6ohn Wallis (Eng.) publish- ii-st record of 'ari effort to. represent ed great work on algebra. an imaginary number graphically. . . 1687 ewton published his Prin Greate'st contribution made cipia. one nian to science. Origin of applied mechanics.

1700 erlin Academy. - 61 -

With the close of the seventeenth century we come to the end with respect to the history of elementary mathematics. By this time the arithmetic of commerce and industry was practically what it is today, and the algebraic content of modern secondary-school texts was practically all in use 53 before the eighteenth century. The general outline which textbooks follow today was devised by Harriot in the seven-

teenth century. ~ubsequent textbooks changed in their me thods of presentation and in their applications to meet the advances in educational theor7 and practice, but t!:eir mathe- 54 matical content remained fairly stable. One outstanding refinement, which revolutionized the whole concept of algebra, was made by G. Peacock in the nineteenth century. He was the first to perceive common al gebra as an abstract hypoth•tico-deductive science of the Buclidean pattern. Following Peacock's lead the British

, - School laid the rational basis for algebra, developing it 55 deductively from postulates. They introduced into algebra the terms commutative, distributive, and associative, and 56 the orientation of material these terms imply. The work of the British School was wholly in accord with the spirit of modern mathematics, one of abstractness and generality. This outline, .which terminates with the close of the seventeenth century, may be considered incomplete without an indication at least that mathematics flowered into its golden age in the twentieth century. A detailed outline of the re- - 62 -

maining two and a half centuries is beyond the scope of this work. However, we may list a few of the great mathematicians of these centuries and leave it to the devotion of the reader to investigate their work. In the eighteenth and nineteenth centuries were the Bernoullts, Euler, De M.oivre, Lagrange, Monge, Laplace, Fourier, Gauss, Cauchy, Cayley, Boole, Ga- lois, Weierstrass, Lobatchewsky, Riemann, Dedekind, Poin- 57 car~, to mention some among many. It seems fitting to close this chapter by quoting E. T. Bell on the development of mathematics after 1800 A.D. As mathematics passed the year 1800 and entered the recent period, there was a steady trend toward increasing abstractness and generality. By the mid dle of the nineteenth century, the spirit of mathe matics had changed so profoundly that even the lead ing mathematicians of the eighteenth century, could they have witnessed the outcome of half a century's progre~ would scarcely have recognized it as math ematics. Another quarter of a century, and it had become almost a disgrace for a first-rank mathema tician to attack a special problem of the kind that wou;l.,d have engaged Euler in much of his work. Ab stractness and generality, directed to the creation of universal methods and inclusive theories, became the order of the day. 58 - 63 -

FOOT-NOT~S FOR CHAPTER II

1. D. E. Smith, History of Mathematics., Vol. 1, p. 37. 2. Sir Banister Fletcher, A History of Architecture, pp.46-54. 3. For pictures of daily life in ancient Egypt suitable for oulletin board display and for a very readable story of Egyptian culture see William C. Hayes, "Daily Life in Ancient Egypt," The National Geographic Magazine, LXXX, (October, 1941), pp. 419-515. 4. Fletcher, op. cit., pp. 11-14. 5. R. C. Archibald, Outline of the History of Mathematics,pJJ.. 6. For pictures of daily life in ancient Greece suitable for bulletin board display ana for a very readable story of Greek culture see Edith Hamilton, "The Greek Way," The National Geographic Magazine, LXXXV, (March, 1944),PJ):"257 - 72. Richard Stillwell, "Greece--the Birthplace of Science and Free Speech," The National Geographic Magazine, LXX.XV, (March, 1944), pp. 273-353. 7. Stillwell, op. cit., p. 279. 8. Fletcher, op. cit., pp. 69-71. 9. For a development of this figure see Alfred Hooper's new book, The River of Mathematica. Henry Holt & Company, Publishers, 257 Fourth Avenue, New York, N. Y., are prin ~1ng copies about 14x20 inches on firm, but not stiff paper, of "The River of Mathematics," a chart .from Hooper's book. 10. "The time curve o.f mathematical productivity is roughly similar to the exponential curve of biologic growth." E. T. Bell, The Development of Mathematics, p. 13. 11. Smith, op. cit., p. 100. 12. Bell, op. cit., p. 68.

13. Ibid., P• 11. 14. Ibid., P• 34. 15. E. w. Pahlow, Man's Great Adventure, pp. 49, 84, 147. 16. Leonard Outhwaite, Unrolling the Map, pp. 1-25. - 64 -

17. Pahlow, op. cit., p. 212. 18. R. C. Archibald, "Babylonian Mathematics with Special Refer·ence to Recent Discoveries," Mathematics Teacher, XX.IX, (May, 1936), PP• 209-219.

19. Bell, 2E._~ cit., P• 85.

20. R. C. Archibald~ Outline of the HiSx>ry of Mathematics, P• 25. 21. Bell, op. cit., p. 79. 22. Pahlow, op. cit., PP• 228-9. 23. Bell, op. cit., PP• 79-86. 24. Smith, op. cit., p. 244. 25. Bell, op. cit., p. 102. 26. Smith, op. cit., pp. 200-65. 27. Ibid., PP• 138-77. 28. For a rather complete oiscussion of Hindu numerals and the number concept, see Tobias Dantzig, Number the Lan guage of Science, New YorkJ The Macmillan Co., 1939, arrl Levi L. Conant, The Number Concept, New Yorks Harcourt, _Bruce, & Co. 29. Archibald, op. cit., p. 24. 30. Lancelot Hogben, Mathematics for the Millions, p. 292. 31. Archibald, loc. cit. 32. Bell, op. cit., p. 95. 33. "The greatest mathematician of the time (early ninth cen tury), and, if. one takes all circumstances into account, one of the greatest of a.Ll times was Al-Khowa.rizmi." F. Cajori, A History of Mathematica, ed. 2, N. Y.,1917. "Their (the Moalems) work was chiefly that of transmis sion, although they developed consideraule originality in algebra and showed some genius in their work on trig onometry. 0 D. E. Smith, History of Mathematics, Boston, 1923-5, 2 vol. "If the work produced (by the Moslems) be compared with that of Greek or modern European writers it is, as a whole, second rate both in quantity and quality." Vera - 65 -

Sanford, A Short History of Mathematica, Boston, 1930. Quoted from Bell, op. cit., p. 94. 34. Bell, op. cit., p. 111. 35. Ibid., P• 91.

36. Smith, op. cit., p. 97. 37. Ibid., p. 143. 38. Pahlow, op. cit., p. 230.

39. J. H. Robinson, Modern Times, pp. 166-180. 40. Pahlow, op. cit., PP• 278, 281. 41. Smith, 02. cit., 'P• 154. 42. Pa.hlow, op. cit., P• 337.

43. ~-, PP• 326-27. 44. l.!:?.!.Q.. , PP• 367-410. .:>mi th, 02. cit., PP• 358-9. 45. Ibid,, P• 360. 46. I. B. Hart, Makers of Science, p. 136. 47. !B.!.£., P• 117. 48. Smith, op. cit., p. 417.

49. Bell, 02• cit., P• 115. 50. Ibid., PP• 115-50. 51. Ibid., pp. v-xi. 52. Lancelot Hogben, Mathematics for the Millions. 53. Smith, op. cit., P• 444.

54. ~., P• 413.

55. Bell, op. cit., PP• 164-5. 56. f.2!£·, P• 550, Chapter, 8, Note 11. - 66 -

57. For the biographies of these men and the mathematics they developed see E. T. Bell, Men of Mathematics. 58. Bell, op. cit., p. 153. CHAPT.b;R III

E.X.'rENSION OF '.fi:O:!; NUMBER dYSTEM - 68 -

INTRODUCTION

As previously stated, the function of the chronological outline is to give an integrated picture of the history. of elementary algebra in its relation to the development of mathematics and of civilization as a whole. In this chapter we shall descend from this over-all view, and snny the de velopment of a particular topic of elementary algebra, the extension of the number· system, in more detail. The story of number runs like a thread through the whole history of both mathematics and al5ebra. Repeatedly in the outline we noted that early cultures were hamper·ed by a poor number symbolism and by the lack of an extended number system. As a consequence progress in the art of calculation was very slow, and in solving equations mathematicians for centuries rejected negative and imaginary roots. We shall take up the study of the extension of the number system at•ita beginning, the number concept of prehistoric peoples, and follow its development to include those numbers found in elementary al gebra.

'rhe story of the development of the number system is 11 not the story of brilliant achievement, heroic deed, or noble sacrifice. It is the story of blind stumbling and chance discovery; of groping in the dark and refusing to ad mit the light. It is a story replete with obscurantism and prejudice, of sound judgment often eclipsed by loyalty to tradition, and of reason long held subservient to custom. - 69 -

1 In short, it is a human story." The beginnings of the number concept may belong more to the domain of arithmetic than of algebra, but a clear understanding of the beginnings of number is important to the com plete development of the topic. The early part of this study may be of practical value for those students of elemen tary algebra who know very little, if anything, about the history and significance of the numbers they use so cormnonly. These students frequently have difficulty with those algebraic problems which presuppose an understanding of such terms as digits, integers, fractions, place-value, and significant figures. The manipulations involved in long division and the extraction of square roots, both in arithmeeic and in algebra, may be more easily explained to stude"nts, once they have grasped the significance of the place-value assigned to 2 number symbols. These two processes are frequently drilled without understanding. Two questions that may be answered here area why were numbers invented and why was the system of real integers ex tended? Numbers were invented to answer such questions as 3 how many ana how long. These answers were in the form of numbers, which grew to the system of real integers, 1, i, 3, 4, 5, ... , as early man felt the need for larger and larger numbers. Many centuries later this number system was gradually extended to include fractions, irrationals, nega- tives, etc. to meet the demands of geometry, commerce, al- 4 gebra, physics, analysis, and the higher arithmetic. - 70 -

In tracing the history of the extension of the number system recourse must be had not only to history, but to archeology as well. History is the study of man through written records,--books, documents, chronicals, codices, papyri, CWleiform tablets, and hieroglyphic inscriptiona. Archeology takes man's reco:r·d back from the dawn of history 5 to the remotest past. To supplement the findings of archeology, anthrop:logists study present and relatively recent tribes, and by analogy form theories concerning the earliest developments of man. There are still aborigines in remote and isola.ted corners of the world today, and there are many extant records written by explorers and missionaries, concerning primitive tribes. By studying the notion of number among such tribes, we may, by analogy, draw some conclusions regarding the beginnings of number prior to the first records of our own ancestors. Such conclusions, by reason of the many ·f.acto:r·s. of error involved, are only general and are not to be accepted as definite and certain.

6 'rlfh NUMBER CONC:b;P'l1 IN PRE-HIS·rORY It has been found that, of all the barbarous tribes, of which studies l:ave been made, there was not a single one that did not show some familiarity with the number concept. A few, however, had not even progressed to the stage of having pure number words in their language, a condition that placed them very close to the lower limit in the scale of civilization. A member 01· such a tribe was able to dis- - 71 -

tinguish between one object and two objects, and between two objects and many objects, but he had no word for two trees •• How the ability to distinguish between one object and two objects evolved, and a description of the primitive state from which it developed, is a matter of conjecture i'or some authorities, and a matter which other·s place out side the limits of scientific investigation. Among aborigines there seems to have been practical methods of enumeration long before words for numbers became a part of their language. If a returning hunter wished to tell anotl~er he had killed three birds, he held up thit"ee fingers to indicate this number fact. He had no word in his language yet to convey the idea of three birds. Other devices for enumeration were the joints of fingers, fingers and toes, sticks, pebbles, shells, notches in sticks, knots on a string, etc. After repeatedly using his fingers to in dicate· a number of objects, a savage, in time, associated the words for· his different fingers with the number of ob jects tallied upon them, and instead of only pointing to the fingers that indicated three birds k!illed he began also to say the words for these fingers. He thus began to form number woras. When a savage fi1·st put objects in a one-to one correspondence with his fingers in an orderly sequence, he did his first counting. Various tribes reached different limits as to how far they could count. Some could count to three, others to seven, etc. In general, primitive man often - 72 -

lost his sense of accuracy after seven, just as children lose their sense of accuracy when they use numbers beyond the limits of their concrete experiences. Though a savage could only count to seven he could still tell whether his flock of fifteen sheep had gained or lost i~ number. The savaee may have kept a bundle of sticks, or a small heap of shells, one stick or shell for each sheep. Taking this bundle of sticks he removed one stick for ever·y sheep. If any sticks remained after each sheep had been accounted for, he had lost some of' his flock. And vice ver·sa, if any sheep remained after· a.11 his sticks had been removed from the bun dle, his flock hllid increased. In a similar manner, we may notice whether a.ny students are absent from a classroom by placing students and desks in a one-to-one correspondence, without knowing exactly just how many students or desks are in the room.

In the course of time, primitive people, it seems, quite naturally stumbled upon the easy method of counting in groups, a convenient device for saving words and for assisting the memory. Various tribes developed various ways of grouping. Some grouped objects by two's (two hands); others by three's (three joints to a finger), by five's (five fingers), by ten's (ten fineers), by twenty's (ten fingers and ten toes). These groupings determined the bases of the respective num ber systems developed. Primitive man, even though he had progressed to the beginnings of a number system, did not acquire the concept - 73 -

·of number until he made the abstraction that three birds~· possessed something in common with three arrows, three men, three trees, etc. When he made this important step we do not know, but it certainly preceded the actual writing of records. It must be remembered that for primitive minds, wholly unused to abstraction, thts process, as well as the . I processes· of building a number scale and a written language, was a long and difficult one, and· was related to the needs - brought on by the gradual development of civilization• As savage tribes gradually changed from a nomadic life to one of living in permanent encampments they develo_ped an agri cultural civilization and became interested iri the progres sion of the seasons. The fertility of both crops and ani- mals were related to the seasons, and careful observations of the heavens were made, not only to insure good crops, but also to set the dates for civic and religious festivals. Seasons and years covered large periods of time, and neces- sitated the invention of some form of permanent record. The beginnings of the earliest number scripts were marks chipped op stqne or wood, or marks _made. in pliant clay, vfhich was

! later baked. It is thought that number scripts preceded scripts for words. Once man had the idea of re,cording numbers, he ap plied the idea to recording the sounds of his language as well. Until alphabets were devised these recordings took the form of ideographs. 74 -

At some stage in his development of the natural number system man, for the first time, performed the simple oper·a- 7 tions of additions, subtraction, multiplication and division. As early man left no records describing the gradual steps by which he slowly passed from the concrete to the abstract, and it is inconceivalbe that he should have, we can only surmise probable situations that led to developing these operations. 'There must have been innumerable occasions for early man to make additions to his property, and through countless ac~ as repeatedly adaing two spears to three spears and counting the total to get five spears, there gradually developed the abstract process of adding two spears to three spears .to get five spears without resorting to counting the total. After man had completely abstracted number and the fundamental operations from concrete objects, he could add two to thr·ee anci.get five, no matter whether he added spears, fish or gods•· In a somewhat similar fashion the subtraction process may have arisen from comparing one's wealth with the wealth of another. How many more skins does he have than I? Or to be more refi.istic, because we are speaking of savages, how many more heads or scalps does this warrior have than r:·

Upon such grim trophies 01· war oi"ten-times uepended an indi vidua.11 s power in the tribe. Division may be considered as a repeated subtraction, and, as such, may have Ci.eve loped from apportioning the spoils of a hunt or a war G.mong the participants. Multiplication, as a r-e:i,;eated audition, may have r·csulted from the practice oi· countin:) in groups. No - 75 -

matter what the occasions were that prompted man to use these t·undamental operations, the process of developing them was one of recognizing, faintly at first and then more clearly through countless repetitions in a variety of situations, a pattern in the way he arrived at answers to his number qµestions. As savage tribes became civilized they developed systems of writing, both f'or numbers and for words, no two of which were alike. We shall investigate brief'ly the symbolism de veloped by a few of the more important early civilizations

for writing the natural numbers, l, 2, 3, 4, 5, •••• Early number symbols were in a constant state of change, being perf'ected through use. To see the weaknesses and short comings of these early systems the reader may contrast the characteristics of these systems with the characteristics of a civilized number system. A civilized number system should: 1) Represent exactly any conceivable quantity. 2) Contain a small number of different symbols.

3) Express large quantities with relatively few figures. 4) Possess simplified mathematical processes. 5) Be adapted to as many laws of natural phenomena as possible, so that computations involving them 8 may be facilitated.

BABYLONIAN NUMBER SYd'.PEM The early Babylonians, the people who settled on the - 76 -

fertile plains between the Tigris anu Euphrates rivers, wrote on a soft clay with a pointed stick, called a stylus. The resulting wedge-shaped. character·s have been called. cu- . 9 neiform writing. A record was made permanent by baking the clay tablet after it had been inscribed. Samples of these tablets have been unearthed by varioµs expeditions to the lands where early Babylonian ci v-ilization flourished, and the key for translating them was wor·ked out about the year 1925. For an example of· such a tablet see picture No. 1, page 77. Otto Neugebauer, an Austrian, has been studying these tablets during recent years and has made discoveries that may show that the Babylonians attained much higher level of mathematical achievement than earlier in- 10 vestigator·s described. The Babylonian number system was both sexagesimal and

deo-imal in character, with authorities d.iffering in their 11 opi~~ons as to which character predominated. The following example, taken from a cuneiform tablet, will clarify 12 the general nature of the Babylonian number symbdB.

"an.son t an-1.6on-l + ... + aJ..60 + a 0 .600 + b_ 1 .60-l + ... + b_m• 60-m, or, nl3 b -m• - 77 -

Picture No. 1

NUMERALS OF TEE 28TH CENTURY B.C. A SUMERICAN TABLET

The numerals at this time were made with the upper end of the scribe's stylus and appear as curved symbols, and as such

can easily be recognized.

Illustration from Breaated1 s Ancient Times. Reproduced from a History of Mathematics, Vol. 1, by D. E. Smith by permission of the copyright holders, ... ,· Ginn and Company. -77R-

NUMERALS OF THE 28TH CENTURY B. C. - 78 -

The Babylonian symbol for 1 was a single upright wedge, (Y); the integers, 2, 3, ••• , 9, were written as two wed ges, three wedges, ••• , nine wedges. The Babylonians used

a special symbol f'or 10, ~ ) • The remaining numbers up to 60 were compounded by making use of these two symbols, the symbol for unity and the symbol for ten. Instead of having a special symbol 1'or 60, the early writers used the single upright wedge. The Babylonians used a place-value system with a single upright wedge representing: 5on, ... , .. . , (1/6o)n. , Translators, consequently found great ambiguity in the writing on these early tablets, liint!l, about the time of the Greeks, the Babylonians invented a special symbol for zero ( i ) , which removed the ambiguity by determing the e xact place a single upright wedge should occupy. Yf f•euld mean that the wedge now stood for 602 with nothing adaed; Yf Yf meant 602with 2 added. In time special symbols were developed for such numbers as 60, 600, 3600, 36000. There are no indications, so far, that the Babylonians made full use of the advantages in herent in the place-value system. Their symbolism was made clumsy by bein overburdened with the repetitions of two sym bols, the Y and the < . Individual writer added variations to overcome the obvious short-comings, but there was no standardized procedure established, other than that determined 14 by bhe nature of the writing tools used. In fact none of

- 80 -

to the numbers represented. A picture of a survey or a chain of one-hundred units represented 100; the symbol for 1000 was, at one time, a lotus flower, of which there were many in Egypt; 100,000 was a tadpole, an animal, which in the adult stage, is noted for laying vast number of eggs; l,000,000 was a man with arms outstretched in astonishment 19 at so great a number. Such large numbers were not frequently used, and symbols used to represent them differed wiaely. (See picture No. 2, page 81.) For present purposes it suf'i'ices to give a few of the ordinary forms of the hieroglyphic numerals. 'rhe ntm.bers from l to 9 wex·e represented by the proper number· of ver tical strokes. Ten had a special symbol,(), aa did. 100 ,(!!, • The numbers fromel..even to ninety-nine were formed by the proper combinations of strokes and n1 s. Thus, ·49 was written; ru 1u lll " ~ 11 f\ Ill 111 11 (\ or r-, ri Ill Sometimes the numbers were read from left to right, and at other times from right to left. For examples of both hie roglyphic and hieratic forms of Egyptian numerals see pie- ture No. 3, page 82. The Egyptians developed a number system based on ten, having special symbols for 10, 100, and the higher powers of 10. Their method of symbolism had all the disadvantages of the Babylonian system, in addition to having the incon- veniences resulting from the lack of a zero and a place- value notation. - 81 -

Picture No. 2

EGYP'.CIAN HThROGLYPHIC NO'J!A'l1ION

Reproduced from A Hiatorz of Elementa!7 Mathematics by Florian CSjori by permis sion of the copyright holders, the Mac millan Company.

- 96 -

such an extent (see Picture No. 6, page 97) that there is little resemblance today between European and American numerals and those in present use in Arabia and India. In this chapter the term Hindu-Arabic will be used to -desig nate the number system that was introduced into Europe in the eleventh century, although in their present form these numbers might just as well. be called European pr Modern 36 numbers. It is difficult to trace the westward route of the .early Hindu numerals- with any degree of certainty because during the Dark Ages in Europe few commercial and scienti- fie works were written, documents that might have used these numberals, and from which historians could have stu died their spread. Authorities have expressed varying opi- nions concerning the introduction of these numerals into 37 38 Europe. In general, two broad theories have been advanced. 1. The numerals were brought to Spain in the eighth and ninth centuries by the Moors or west Arabs, who in turn received them from the Hindus via the east Arabs. From Spain the numerals spread through wes tern Europe. 2. The theory advanced by Woepcke is that the numerals were already in Spain when the Moors got there, having.readhed the Mediterranean coast in the fifth and sixth centuries via the coastal trade routes and the Nee-Pythagoreans. Following the second theory we may say tha.t from India ·the Hindu numerals, without the zero-, spread via the trade routes to Mesopotamia and Alexandria.. They had reached Alexandria by the fifth century A.D., and·were taught in the monastic schools in Mesopotamia in the middle of the seventh

- 135 -

For the sake of an orderly treatment of the use of his torical materials in the classroom, we may say that, in gene ral., such materials may be used either by the teacher alone, or by the teacher in cooperation with the students. A tea cher has ample opportunity to draw upon his knowledge of the historical background of his subject in answering the ques tions raieed by members of the class, in giving historical overviews of topics, in maintaining interest, and in the selection and use of film slides, f'ilm rolls, and movies. In cooperation with students a teacher may wish to create a mathematical atmosphere in the classroom or develop the bul letin board as an ef'fective teaching aid. Such activities as mathematics clubs, progrmas for school assemblies, math ematical exhibits, and field trips to museums and other places of mathematical interest call, likewise, for teacher student cooperation. Students frequently, ol their own accord, raise questions which clearly indicate that in their own minds they are seek ing understandings broader than those of mere subject matten. The following questions are typical of' almost any class stu dying algebra, and their answers are largely historical. W.b.y should we study algebra? Has this study ever benefited any one? Why lirst studied this type of. equation? ~hy do we write a•a•a• as a3? Where did these rules come from? Why are there so many different methods of indicating multipli cation? Why have we followed these particular rules in long division?