led without delay to perceive that the lesson learned today is little more than the cultivated ground out of which is to grow the lesson of to-morrow.

It is a fatal error, only too common, to start a child to study where that which he is asked to learn is out of touch with that which he already knows. Pupils should be taught very early to keep an accurate separation of the known from the unknown, and "to be careful not to stamp a thing as known" until they have fully mastered it in all its relations to that which they know, and have done so "in that way which conscience calls honest.”

The following Elementary Treatise on Arithmetic has been prepared with the view of presenting to both teacher and pupil a thoroughly systematic and gently graded scheme in which they may together make daily progress in scientific knowledge of the subject, and by a mutual interest in the work gather by diligence many of the best fruits of industry.

Nothing, from beginning to end, has been written as mere verbiage, undeserving of attention. Every word has a measure of significance, and every sentence in the book is there for the single purpose of being understood.

J. M. R.

GENERAL SUGGESTIONS.

PART I. of this book presupposes the occasional necessity of introducing pupils to the first rudiments of the subject.

In developing the idea of number, and teaching the simple numerical operations, real objects are greatly to be preferred to pictures, however artistic and striking they may be. But what is to be preferred to all other agencies, whether pictures or real objects, or even the book itself, is the voice and action of the live and intelligent teacher, without which little of educational value is ever accomplished in school. Object lessons, however, are of great value in illustrating and impressing the teacher's meaning; but no objects should be used except those of simple form and construction, lest the mind of the pupil be diverted by them from the primary object in view, namely, the inculcation of the idea of number and of numerical combinations.

When objects are employed, the youthful pupil should be allowed to take them in his own hands and give proof of knowledge gained by showing without help how and what they explain. This will be to his liking, and liking is a supreme element of learning.

The successful teacher is not he who does both his own and the pupil's work, but he who best directs the pupil's

activities, leads him to love learning, and to overcome difficulties by his own efforts.

In reciting, pupils should be required to give their answers in complete sentences, and that, too, without hesitation or counting. They should be able to say "4 and 3 are 7," or "4 less 3 equals 1," as promptly and with as little apparent effort as they would spell a word of three letters. In sight exercises, where rapidity is the object sought, results alone should be given; thus: """ one."

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seven,"

The use of concert exercises should not be constant, but only occasional, as a ready means to give diversity or to revive flagging interest. As far as possible, the members of a class should be drilled individually, for each has a separate and distinct individuality that demands and must receive from the teacher the carefulest recognition. Every exercise should be made so interesting as to engage the undivided attention of every pupil, and to effect this they must of necessity be lively, varied, and above all brief. The very best judgment of the teacher is demanded here. The processes of Addition and Subtraction. are so intimately related that they should be taught at first together. In like manner should be taught the closely related operations of Multiplication and Division.

All written work should be as neat and well arranged as the pupils are capable of doing it. Ill-formed figures and careless arrangement are fruitful sources of error in results.

Pupils while in school should be kept constantly employed. They must do if they would learn. Idleness should not be tolerated for a moment. The forward