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continuity of the whole. Such topics have been accompanied by the star (*).

Considering the various chapters in some detail, Chapter I acquaints the pupil with certain elementary notions that are central in algebra, more especially that of the literal number and the simple equation. The development at this point is easy and natural, being based upon familiar simple principles from arithmetic, and the whole is abundantly supplied with problems of a kind that will instinctively appeal to the average beginner, and thus invite his early interest in the subject. Chapter II, where negative number first appears, contains an unusually large variety of illustrations tending to bring out the full significance of such numbers. Chapter IV, entitled Multiplication and Special Cases of Factoring, departs slightly from the usual procedure in that it develops the elements of factoring along with those of multiplication instead of delaying the entire subject of factoring for a later chapter. In this way the two subjects (which are but the reverse of each other) are seen from the beginning in their mutual relations. However, it is only the simpler types of factoring that are taken up at this early stage, the more difficult ones being reserved for the following chapter. The result of this arrangement, as shown by experience, is that factoring, having thus had an early and natural beginning in connection with multiplication, becomes less isolated and consequently is the more readily grasped by the beginner.

Passing to the later chapters, it may be observed that in the treatment of surds, radicals, and roots (Chapters XIV and XV) the pupil is taught the use of tables — a topic heretofore quite neglected in elementary algebra, yet one of increasing importance owing to the number of pupils that pass from our high schools directly into technical pursuits.

A word should perhaps be said here regarding the treatment of graphical methods that the book presents. The authors believe that while such methods may be (and indeed often are) introduced into algebra at a very early stage, yet the most reliable experience indicates that the pupil has as much as he can reasonably be expected to do at the very outset if he masters merely the meaning and technique of the literal number and simple equation. This much he can do immediately from his knowledge of arithmetic, while graphical interpretation introduced at this stage is apt to seem artificial. Graphics are therefore postponed until Chapter XII, at which time the pupil has a substantial groundwork in algebraic facts and reasoning.

Volume II begins with a systematic review of the fundamental processes of algebra. This is followed by a more extended treatment of certain of the topics in Volume I and this in turn by chapters upon a number of the more advanced topics required for entrance into our best colleges and technical schools.

In conclusion the authors would here acknowledge their indebtedness to various other texts, especially the recent treatises of Godfrey and Siddons appearing in England, and to various current discussions and articles such as have appeared from time to time in School SCIENCE AND MATHEMATICS and in the AMERICAN MATHEMATICAL MONTHLY. The authors are also indebted to Professor L. C. Karpinski for the sketch of the history of algebra appearing at the opening of the book, and they would here express their gratitude to certain friends who have kindly examined the manuscript and proof sheets and offered timely suggestions and criticisms — in particular to Professor E. R. Hedrick.

WALTER B. FORD.
CHARLES AMMERMAN.

FOREWORD TO THE PUPIL

ALGEBRA is not a difficult subject, but you must remember to read carefully what the book says. Every sentence is important, and if you fail to understand one sentence you will be quite likely to have difficulty from that point on through the later pages. There are times perhaps when, after repeated efforts, you cannot understand at a certain point — at such times you should freely express your difficulty to the teacher, who will explain the meaning and help you.

Remember also that it is best to examine carefully the solution of the problems worked out in the book before you attempt to do other problems like them. For example, before attempting the problems on page 138 you should have examined carefully those of a similar kind that are completely worked out on page 137.

HISTORICAL INTRODUCTION

By Louis C. KARPINSKI ALGEBRA had its beginnings in a very 'remote period of history, probably not less than 3500 years ago. We know, in fact, that at about that time a certain Egyptian named Ahmes (pronounced A'-mes) wrote a mathematical text-book in which he proposed several problems containing equations.* For example, one of the problems reads as follows : “An unknown and its seventh make 19. What is it?” In solving this, Ahmes did not use a letter to represent the unknown, as we would naturally do now, but the steps he followed were nevertheless essentially algebraic. Other Egyptian texts containing similar problems are preserved to-day in the museums of Paris and Berlin.

During the era of Greek civilization and culture, which followed that of ancient Egypt, algebra made but little progress because the Greeks were interested chiefly in geometry. However, Euclid (U'-klid), who lived about 300 B.C., and Archimedes (Ark-i-mē’des), who lived at Syracuse on the island of Sicily about 250 B.C., both of whom were great mathematicians of this period, could solve first-degree equations and even quadratics, but they used geometric methods instead of the simpler algebraic methods which we now have.

* Ahmes' work is still preserved and is to-day in the British Museum. Like all manuscripts of such antiquity, it is written upon papyrus, or paper made by pasting together long thin strips cut from the papyrus plant.

Following the Greeks, the Romans had little or no influence upon the development of mathematics. In fact, we find little progress in algebra except in India until about 800 A.D., at which time the leading scientists of the world were among the Arabs. In particular, the Arab mathematicians familiarized themselves with the writings of the Greeks and Hindus on algebra, and made advances over what they had thus received. As a result, the first systematic treatise on algebra appeared, being written by an Arab named Mohammed ibn Musa al Khowarizmi. Here it is shown how algebra may be applied to the solution of certain geometric problems and to certain engineering questions. It may be remarked that the same author also wrote a famous treatise on arithmetic, which was long used. This work brought into Europe for the first time the numeral system (called the Hindu-Arabic system) which we use to-day and which employs the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 instead of earlier symbols, such as the I, V, X, C, and M used by the Romans.

By the year 1200 A.D. many of the Arabic works dealing with science had been translated into Latin and it was chiefly through such translations that algebra became known to the Europeans of the Middle Ages.

Modern algebra may be said to have begun with the French statesman Viète (Ve-ěte', 1540–1603), who was the first to use letters to represent known and unknown quantities. He used vowels to represent unknowns and consonants to represent knowns. Our common use of x, y, and 2 to represent unknowns arose with the great French mathematician and philosopher Descartes (Dā-cart', 1596-1650) and was adopted soon afterward by the great English mathematician and astronomer Sir Isaac Newton (1642–1727) and by the great German philosopher and mathematician Leibnitz

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