the first numbers, they are able to make small calculations. And this we see them do every day about their play-things, and about their little affairs which they are called upon to attend to. The fondness which children usually manifest for these exercises, and the facility with which they perform them, seem to indicate that the science of numbers, to a certain extent, should be among the first lessons taught to them.

"To succeed in this, however, it is necessary rather to furnish occasions for them to exercise their own skill in performing examples, than to give them rules.”

He goes on to speak of the plan of the work, as follows: [The italics in these extracts are our own. n.]

"Every combination commences with practical examples. Care has been taken to select such as will aptly illustrate the combinations and assist the imagination of the pupil in performing it. . The examples are to be performed in the mind or by means of sensible objects, such as beans, nuts, etc. The pupil should first perform the examples in his own way, and then be made to observe and tell how he did them, and why he did them so."

Again, in criticising the ordinary way of teaching arithmetic, he says:

"The pupil, when he commences arithmetic, is presented with a set of abstract numbers, written with figures, and so large that he has not the least conception of them even when expressed in words. From these he is expected to learn what the figures signify, and what is meant by addition, subtraction, multiplication, and division; and at the same time how to perform these operations with figures. The consequence is, that he learns only one of all these things, and that is how to perform these operations on figures. He can perhaps translate the figures into words; but this is useless, since he does not understand the words themselves. Of the effect produced by the four fundamental operations he has not the least conception.

"The common method, therefore, entirely reverses the natural process; for the pupil is expected to learn general principles before he has obtained the ideas of which they are composed."

We should be glad to see Warren Colburn's original preface published by itself, as an educational tract, and put into the hands of every teacher in the country. A friend of education, desirous of doing the greatest possible good with a small expenditure, could hardly do better than to spend the money in printing and circulating this admirable little treatise on elementary instruction in numbers.

The introduction, by Geo. B. Emerson, contains some very valuable suggestions in regard to the use of the book in our schools. We can not refrain from quoting a few sentences from it, in the hope that they may lead you to read the whole :

"It is strictly a mental arithmetic; and if faithfully used in the way intended by the author, it evolves from the mind of the learner himself, in a perfectly easy and natural manner, a knowledge of the principles of arithmetic, and the power of solving, mentally and almost instantly, every question likely to occur in the every-day business of common life.

"It can be well taught only by a teacher who perfectly understands it, and who

knows how to teach. Such a teacher will not allow the lesson to be previously studied by the pupil. Each section is intended to teach some one process up to a certain point. If, in the course of the section, questions occur which the class can not readily solve without previous study, the teacher has only to interpose, at the point where the class fails, or begins to fail, additional questions of the same kind, somewhat easier than those in the book. If, at the end of the section, the class be not perfectly ready in the solution of the questions, the teacher ought to go over the section again with the class, or to add, at the end of the section, a sufficient number of similar questions to render the solution easy and instanta

neous.

"By allowing the class to study the lesson beforehand, not only is much time lost, but the exercise is turned into a poor sort of mechanical process not much better than the common ciphering. Its mental character ceases almost entirely."

We believe that Mr. Emerson is right in claiming that Colburn's 'First Lessons' is as nearly perfect as any human work well can be. And yet we have scores of Mental Arithmetics' intended to supersede it.

A few of these are the result of an honest endeavor to write a better book. Their authors could not see, as Mr. Emerson expresses it, 'how really profound and comprehensive' Colburn's is, and in substituting their own crude notions for the philosophic completeness of the original, they have only paraded their blindness and ignorance.

It would be worth the while, if we had the time, to illustrate this by comparing portions of one or two of these 'improved' mental arithmetics with Colburn's treatment of the same subjects. We should like to illustrate by comparison of the books, but we have already extended this article too far.

Massachusetts Teacher.

PRIMARY AND GRAMMAR TEACHERS.

In the history of our public schools no subject has received from the friends of education more attention than Primary Instruction is receiving at the present day. In the majority of the reports of Superintendents of schools which have been published during the last two years this subject has received special notice. The principal topics which have been considered in these articles are- - the importance of primary instruction; the necessity of the best teachers for primary schools; teachers of primary schools should be the best paid. That primary instruction is not important I have no intention of asserting; but that it is not of so much greater importance than grammar instruction I do believe, and shall attempt to demonstrate.

In the primary school children learn their letters; in the grammar school they learn the first principles in grammar and arithmetic. Upon which foundation is there to be the most building? Will the manner in which those children learn the alphabet exert a greater influence upon their future course of study than that in which they learn to study the sciences? Why are so many people unable to explain the inverting of the divisor in division of fractions? Is it because they are not taught the alphabet in the right way? or is it because the foundation in arithmetic was not well laid?

It is said that the best teacher, should be in the primary department, and if mediocrity must preside at the teacher's desk, let it be in the grammar department. By best teachers I understand teachers who possess the faculty of teaching in a pleasing way, of making crooked things in learning straight, and of imparting life and vivacity to scholars. Does it need any more tact to teach a child his letters than to teach him to write numbers? Does it require any more tact to teach a child to draw than to write? Vivacity is needed in a primary school. Does it require any more energy to interest children in their tasks when they have scarcely thought of the work than it does to interest children who have worked upon some principle during the previous evening, failed to comprehend it, and finally enter the class with the idea that it is dull and hard, and they can not understand it? If there is any thing within the province of the primary school which requires more tact than it does to make children believe that Case is a subject which they can master, and make lessons in disposing of substantives interesting, I would like to learn what it is.

From the facts that primary instruction is the most important part of instruction, and that the best teachers should be in the primary department, it is deduced that primary teachers should be the best paid. With so much favor has this been received, that in Chicago the teachers in the lowest two grades receive ten dollars more a year than the same teachers would receive if in a higher grade. Is this just? Does the primary teacher exhaust her bodily strength more than the grammar teacher? Is it any easier to make children from nine to twelve years of age understand arithmetic, grammar, and mathematical geography, than to draw out the ideas of little children in an object lesson? When a primary teacher has dismissed the last child she can leave the room, lock school-duties in, and take the recreation of mind and body which is essential for all. Let me ask, How can the grammar teacher leave school behind when she has a package of examination papers in her hand? You say "omit the examinations." No, that can not be done, for experience has taught us that constant written exercises aid

the child more in the formation of the habit of speaking clearly and concisely than any other plan which has been adopted. Written spelling-lessons must be corrected, Monthly Reports to Parents' must be prepared at the first of every month. While the grammar teacher, possessed of education and talent equal to that of her friend in the primary department, must spend her evenings in working upon schoolpapers, her friend has that time to devote to reading and other studies for improvement. Yet the one who has the leisure time receives the most salary, although she works no harder in school and works less out of school.

In writing this article I have attempted to be unprejudiced, and have merely stated what I think to be true. Viewing in the light in which I have attempted to present it the relative importance of the two classes of teachers and the amount of work performed by them, I think I speak the minds of many grammar teachers in saying that manifest injustice is being done them. They are being placed among the second-class teachers, and not permitted to take the rank which they, by the aid of experience and talent, will be fitted to take. Superintendents of Schools who advise, and Boards of Education who decree, that primary teachers shall receive the greater compensation would do well to think more carefully before so doing, lest they check the less ambitious grammar teachers, and cause the truly faithful ones to feel that they are unappreciated.

N. E. F.

MATHEMATICAL DEPARTMENT.

CONDUCTED BY S. H. WHITE, OF CHICAGO.

A WORD TO OUR PATRONS--In taking charge of the Mathematical Department of the Teacher, it shall be our aim so to conduct it that it may afford the greatest assistance to those in the profession. This is the teachers' journal, devoted exclusively to their work; and the more materially it aids them, the more completely it answers its mission. The effort will be made to make it subserve the greatest good to the greatest number, not confining our attention to the higher departments of mathematics alone, a course which would make it more attractive to a few, but which would deprive it of interest to the many.

With this view, suggestions will from time to time be made concerning methods of instruction in numbers, which, it is hoped, may aid younger teachers and, perhaps, furnish food for thought to others. Many of the examples will be within the ability of pupils in the highest classes in many of our schools, so that teachers may use them in testing the real strength of their pupils, while it will be an encouragement to the latter to present systematic and neatly-arranged solutions for publication. Any such solutions by their pupils which teachers may send in (they being satisfied that the work is the pupil's own) will be published or credit given therefor.

Former correspondents with this Department of the Teacher are requested to continue their contributions; and teachers and our friends generally are invited to make this their own paper by sending questions for solution, mathematical queries, or other matter belonging to this Department.

S. H. WHITE.

PROBLEMS.-68. A cellar is 26 feet long, 18 feet wide, and 7 feet deep. How much earth must be removed to make it 28 feet long, 21 feet wide, and 8 feet deep?

69. Divide $135 between three men, A, B, and C, so that one-half of A's shall be equal to two-thirds of B's, or three-fourths of C's.

70. A boy bought a slate, a book, and a pen, for 80 cents. The pen cost 10 cents, and the book cost 12 times as much as the slate and pen. How much did each cost?

71. A tree 150 feet high, standing on a hill-side, was so broken by a wind that the broken piece rested on the stump and reached down the hill 35 feet from the base of the tree, and the distance in a horizontal line from the base to the part broken off was 20 feet. Where did the tree break?

M. J. V.

72. Given (x+y)2+(x+y)=30, x-y-1, to find the values of x and y.

M. J. V.

THE ARABIC NUMERALS, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, are in their nature hieroglyphics; that is, marks for ideas, and not for words. With a single exception, these figures were, without doubt, in their origin, forms composed of straight lines of equal lengths, the number of straight lines contained in each figure corresponding to the number represented by that figure. Those original forms were gradually modified by usage into the forms now used.

What was the original form of each of the Arabic figures?

The above question was originally dropped into the query-box at one of the county institutes of the state.

The Arabs derived the denary system of numbers from the Hindoos, and it dates as far back as the Sanscrit. The original characters