less, combine in each book the theory and the exercises for practice, because this is the English and American custom, giving in our algebras a few pages of theory followed by a large number of exercises. The Continental plan, however, inclines decidedly toward the separation of the book of exercises from the book on the theory, thus allowing frequent changes of the former. It is doubtful, however, if the plan will find any favor in America, its advantages being outweighed by certain undesirable features. There is, perhaps, more chance for the adoption of the plan of incorporating the necessary arithmetic, algebra, and geometry for two or three grades into a single book, a plan followed by Holzmüller with much success.

1 An interesting set of statistics with respect to German text-books is given by J. W. A. Young Hoffmann's Zeitschrift, (X. Jahrg. (1898), P. 410, under the title, Zur mathematischen Lehrbücherfrage.

CHAPTER VIII

TYPICAL PARTS OF ALGEBRA

Outline While it is not worth while in a work of this kind to enter into commonplace explanations of matters which every text-book makes more or less lucid, it may be of value to call attention to certain topics that are somewhat neglected by the ordinary run of classroom manuals. The teacher is dependent upon his text-book for most of his exercises, since the dictation of any considerable number is a waste of time. He is likewise dependent upon the book for much of the theory, since economy of time and of students' effort requires him to follow the text unless there is some unusual reason for departing from it. But he is not dependent upon the book for the sequence of topics, nor for all of the theory, nor for all of his problems; neither is he precluded from creating all the interest possible, and introducing a flood of light, through his superior knowledge of the subject. For this reason this chapter is written, that it may add to the teacher's interest by throwing some light upon a few typical portions, and may suggest thereby some improved methods of treating the entire subject.

Definitions - The policy of learning any considerable number of definitions at the beginning of a new subject of study has already been discussed in Chapter II. The idea is always of vastly more impor tance than the memorized statement. At the same time there is much danger from the inexact definitions to be found in many text-books, a danger all the greater because of the pretensions of the science to be exact, and because there will always be found teachers who believe it their duty to burn the definitions indelibly into the mind.

Whether the definitions are learned verbatim or not, the teacher at least will need to know whether they are correct. For this purpose he will find little assistance from other elementary school-books. He will need to resort to such works as Chrystal," as Oliver, Wait, and Jones, or as Fisher and Schwatt 8 in English, as Bourlet 4 in French, as the convenient little handbooks of the Sammlung Göschen 6 or the new Sammlung Schubert 6 in German, and Pincherle's little Italian handbooks.?

A few illustrations of the general weakness of the common run of definitions may be of service in the way of leading teachers to a more critical examination of such statements.

The usual definition of degree of a monomial is so loosely stated that the beginner thinks and continues to think of 3 r*x8 as of the fifth degree, which it is in a and x; but for the purposes of algebra, especially in dealing with equations, it is quite as often considered as of the third degree in x, a distinction usually ignored until the student, after much stumbling, comes upon it.

A square root is usually defined as one of the two equal factors of an expression, although the student is taught, almost at the same time, that the expression of which he is extracting the square root has no two equal factors. E.g., he speaks of the square root of z% + 1, and yet says that té + 1 is prime.

Even so simple a concept as that of equation is usually defined in a fashion entirely inexpressive of the present algebraic meaning. Some books follow an ancient practice of avoiding the difficulty by introducing the expression “equation of condition," and never referring to it again! In the algebra of to-day an equation is an equality which exists only for particular values of certain letters called the

notes to the new edition of De Morgan's work, On the Study of Mathematics, Chicago, 1898, p. 187.

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unknown quantities. As the term is used by alge. braists of the present time, 2 +3=5 is not an equation strictly speaking, although it expresses equality ; neither is a + b = b + a2, although it is an identity. An equation, as the word is now used, always contains an unknown quantity."

The term “axiom " is subject to similar abuse. No mathematician now defines it as "a self-evident truth," and no psychology would now sanction such an unscientific statement. Algebraists, those who make the science to-day, agree that an axiom is merely a general statement so commonly accepted as to be taken for granted, and a statement which needs to be considered with care in the light of the modern advancement of the science. For example, no student who thinks would say that it is “self-evident” that “like roots of equals are equal." If 4 = 4, it is not "self-evident” that a square root of 4 equals a square root of 4, for +2 does not equal – 2.

Again, of what value is it to a pupil to learn the ordinary definition of addition ? Text-books commonly say, in substance, that the process of uniting two or more expressions in a single expression is called addition; but what is meant by this “uniting"? Either the defin nition would better be omitted, or it would better have some approach to scientific accuracy; the choice of