is asked; the answer to what is asked being implied in what is told. The chief activity of the child, in the solu tion of true problems, is, first, the analyzing of given con ditions, and second, the performing of certain operations. The former of these activities should never become mechanical; the latter should pass into the mechanical stage as early as possible.

The study of advanced arithmetic differs in certain essential particulars from all other studies in the commonschool course. The thought lacks the continuity of history or geography. The work consists of a series of efforts that are more or less distinct. Each problem stands alone in its statement; but the operations involved in it fall under certain general types. Arithmetic, consequently, requires a constant dealing with both particular and general notions. These general processes may be discovered in illustrative problems, and the possible number of them be absolutely exhausted. When this has been done, the "case" arrangement of problems should cease, and the pupils should be permitted to perform that analytic activity that discovers conditions in a problem, and that synthetic activity that unites it to its proper class. The problems in this book have been prepared with this thought in mind.

Another phase of arithmetic work should be clearly appreciated. It surpasses all other studies in the lower grades in the number of its generalizations and the ease with which they are made. The generally accepted principles that instruction should proceed from the concrete to the abstract, and from the individual to the general, have constant application. But the abstract should be the result of a conscious abstraction, and the general

should be a conscious generalization. As illustrating this thought, and showing how the general is reached from the individual, particular attention is called to the method of deriving rules from processes. The formulation and statement of a rule is a generalization.

It is a fundamental principle of true teaching, that whatever is done by the pupil shall be accomplished through his conscious, personal effort, and that whatever his acquisition, it shall be consciously his own, not his in the memory alone, which is the same as his in the book, but his vitally and substantially, as the blood in his veins or the innate ideas of right and wrong. Such knowledge is of a rooted and growing order that gives satisfaction and power to its possessor.

To suggest and stimulate such teaching, and to secure such growth, many questions are asked and many directions are given, in this book, which are designed to throw the pupil upon his own resources. The questions cannot be answered by any statements found on the pages, -no need to tax the memory for words and phrases; yet all the questions and directions are simple and easily to be answered by the pupil, if he has thought his way clearly.

Too great emphasis cannot be given to the statement already made, that the work of analyzing should never become mechanical. While there is value in concise and definite formulas, there is infinitely greater value in freedom. The mind should be free to discover conditions, relations, and sequences; it should be as free in stating conclusions and results; but this freedom cannot exist if the reasoning is compelled, per force, to follow a memorized formula. The forms of analyses given in the fol

lowing pages are presented as models to be studied and mastered, but not to be memorized. If the pupil, in his own way and in different words, shall clearly present the .steps of reasoning and draw the correct conclusion, his work should be approved.

An additional suggestion must suffice. The teacher's knowledge of the subject should be organic. Arithmetic should be recognized as a science that is deduced from the idea of addition. When the so-called fundamental processes have been mastered, little remains but repetition. The fraction differs from the integer in that it introduces a double unity. The decimal fraction differs from the common fraction in the method of expressing its denominator. Percentage is "a case" in decimal fractions. Compound numbers differ from simple numbers because of the introduction of variable scales, etc., etc. As the power to generalize relieves the mind from the overwhelming burden of a countless multitude of individuals, so each new step in advance is easily held if correllated with the fundamental ideas.

J. W. C.