Working by Rules.

I am glad to learn that the rule for dividing a trapezoid, as published in the JOURNAL for February, was intended for a practical rule for carpenters rather than for the solution of the accompanying problem as a school exercise. I was about to write an article for the JOURNAL, on the too common method of solving problems by rules instead of reasons, when the "Plank Problem" met my eye. Seeing at a glance that there was an error somewhere of nearly a square inch in the sum of the two halves, I attributed it to the rule, relying on the known accuracy of the computer for correct work.

The rule is right, and the mistake was in the work. If, however, the reasons for the rule are so obscure as to make enquiry for them proper in the JOURNAL, there is a "better way" to solve the problem as a school-room exercise. Before stating the method I prefer, I wish to disclaim any but the most respectful reference to Principal Pendleton, whose mathematical abilities are well known to the readers of the JOURNAL. His solutions in the April number are excellent. The reader will bear in mind that the advantages claimed for the analytical solution are for their educational value as mental exercises. In the shop or lumber yard, the positive rule, the tables of a "Ready Reckoner," or the slide-rule might be better.

The truths that the areas of similar figures are as the of squares their similar sides and that the area of a triangle is expressed by the product of the base by half the altitude, are not rules.

Problem. To find the length of a line parallel to the base of a trapezoid that shall divide the figure into two equal trapezoids. The question proposed was to so divide a plank, 12 inches wide at one end, 18 inches at the other, and 12 feet long.

Solution. (1). The average width of the plank is 15 inches. (2). The area of the plank is the product of the length by the width, or 15 square feet, and half the area is 71⁄2 square feet.

(3). If the edges of the tapering plank are produced till they meet, they will form the sides of a triangle having for its base the wide end of the plank. The altitude of this triangle will be 36 feet, because the sides, which are three half feet apart at the base, taper one half foot in a length of 12 feet, and hence will taper three half feet in a length of three times 12 feet. The area of a triangle having a base of a foot and a half and an altitude of 36 feet, is 27 square feet.

(4). Suppose 71⁄2 square feet, half the area of the plank, to be taken from the triangle of 27 feet, a triangle will remain of 192 square feet. Hence we have two similar triangles whose areas are 27 and 191⁄2 square feet, and the base of the larger is a foot and a half.

(5). The areas of these triangles are as the squares of their bases, and the base of the smaller is the line required by the question. Hence the proportion 27:19.5:: 2.25 (square of base): Answer squared, or 1.625. The square root of 1.625 is 1.27475 (nearly). Mr. Pendleton's rule gives the same result. The "better way" is the plainer way for the learner to comprehend. The rule may be convenient, but to depend on rules to solve problems in a text-book is to lose the educational advantages they were intended to afford. If the Apostle advised to "sing with the understanding also," we may confidently advise to solve problems with the understanding also, and all teachers know that ready-made rules tell what and how to do, rather than why.

The second way of solving the problems in the April JOURNAL is better than the first, but it is not the usual way. The problems

were

1. How long a lead pipe of one inch external diameter and onefourth of an inch thick, can be made from a solid sphere of lead one foot in diameter? No allowance to be made for waste or change of density.

2. How long a wire .or of an inch in diameter can be made from the lead pipe, as above named?

The capital necessary, for the learner to engage in the business of solving such questions, should include the following truths: (a). Any sphere is two-thirds of a circumscribing cylinder. (b). Lengths of cylinders of equal volume are inversely as the squares of their diam

eters.

I would solve the problems thus:

The sphere of one foot diameter is equivalent to a cylinder of the same diameter and two-thirds of a foot long, or to a cylinder one inch in diameter and 144 times two-thirds of a foot long, which is 96 feet. The question requires the length of a pipe or hollow cylinder with a bore half an inch in diameter. The section of the bore is one-fourth the area of the cylinder an inch in diameter, and hence the length of the pipe will be four times the length of the solid cylinder. Hence a pipe 128 feet long is equivalent in volume to a

solid cylinder 96 feet long (of the diameters named), or to a sphere one foot in diameter. The cylinder 96 feet long and one inch in diameter, is equivalent to a wire having a diameter of one-hundredth of an inch and a length of 100 times 100 times 96 feet, or 960,000 feet.

It has long been the practice in the best schools and colleges for students to spend years in the study of Geometry, and to demonstrate the propositions which constitute the basis of practical calculations in navigation, surveying, engineering, &c. This is right and proper, but it is not wise for students to neglect to use the truths established by demonstration and to use in their stead rules or formulas that are in any way obscure.

Mr. Pendleton's ingenious solutions in the April JOURNAL show that there is no danger of his pupils becoming "rule-bound," but observation shows that rule-bound workers are not extinct.

Norfolk, Va.

N. B. WEBster.

Suggestions to Teachers of History.

By MARGARET COMPTON, State Normal School, Edinboro.

Reconcile yourself in the outset to the fact that the brightest pupil, under the best teacher, can learn, so as to retain, but a small fraction of the vast accumulation of names, dates and facts which constitute the framework of history. It is better to seize a little and hold it rather than grasp at much and let all escape.

Before beginning the study of any epoch, therefore, select carefully three or four leading events, the dates of which you will require your pupils to remember accurately. As these are introduced in successive lessons, explain their importance, showing that the other incidents group themselves about these naturally. Review daily as many of these points as have been previously learned, varying the manner of presenting them frequently. For example, they may be assigned as topics for oral or written recitation, either to particular pupils or to the class. Questions may be framed to draw from the class the desired facts. The teacher may give dates, requiring the pupil to supply the corresponding facts promptly, or vice versa. After the class has completed an epoch, require the pupils to outline it, giving the dates of the selected events only, arranging the others in proper sequence. Whenever this epoch is properly re

viewed, insist on the omission of all dates but those designated. The reason for this is, that the few dates you wish to impress upon the mind will not stand up conspicuously, however often repeated, if others are repeated equally often. As shadows are necessary to bring out the lights in a picture, so nothing but the judicious suppression of less important details will insure a proper degree of attention to essentials.

Teach events in their relation to each other. Help your pupils to realize that history is a growth; that though the laws of its development are imperfectly known, they are as certain in their action as those of the material world. Make clear to them the difference between history and a mere collection of historical facts, arranged chronologically. Disabuse their minds of the idea that a vast store of facts, with no knowledge of their connection, is a valuable mental possession.

Cultivate in them a habit of tracing effects to their causes, and of inferring results from known conditions. Lead them to expect a why after every what. Call for a statement of the probable conse⚫ quences of actions or events under consideration, before the actual results have been learned, and then compare the suppositions with the facts. When these disagree, seek for the errors of judgment which occasion the difference. By thus inducing them to reflect on what they study, you prepare your pupils to make practical use of their knowledge in future in discharge of their duties as citizensthe main object of introducing the study of history into the public schools.

Encourage a search of whatever papers and books are accessible for additional information on any topics studied. Have the results reported in class, always accompanied by a statement of the source. Inspire your pupils with a curiosity that will lead to much question-asking on their part. Do not answer questions at once always. Refer the class to places where an answer may be found, if the subject is of importance and information within their reach. If, as will sometimes happen, questions arise as to which neither you nor your class can at once find an answer, note them down, and read over the list occasionally. If these questions are written neatly on the margin of the text-books opposite the topics which suggested them, and the answers, with the place where found, also recorded with them as soon as possible, it will serve to keep the question fresh in the student's mind while yet unanswered, and will increase the value of the text-book for reference afterward.

Require
Connect

Take a good newspaper and use it frequently in class. your pupils to keep themselves posted on current events. these, whenever possible, with facts they have learned in their lessons. For example, the Civil Service Reform Bill, now before Congress, suggests the opening of Jackson's administration. One recommendation in President Arthur's message recalls the Joint Electoral Commission. Anniversary celebrations of historical events furnish fine opportunities for lively history lessons.

Make the class familiar with the officers of the General Government and their duties.

Aim to instil in their minds a just pride in our country's present position and future possibilities, and a true sense of what constitutes national honor or national shame.

Make a point of acquainting your pupils, in the course of instruc tion, with the names of a few really excellent histories and biographies, which it shall be their ambition to read and possess.

Estimate your success by the promise your pupils give of being thoughtful and interested readers of history after you leave them, rather than by the percentage of correct answers in a difficult examination at the close of the term.-Louisiana Journal of Education.

School Government.

The maxim that "the best government is that which governs least" applies to the little communities of our schools as well as to the large communities of nations. Our best governed schools are those whose time is so wholly occupied by attention to varied and interesting school work that there is no thought of "governing" or "being governed."

The fundamental principle, that the activities of child nature must be turned into constant, engrossing and varied work, or they will develop into restlessness and disorder, is becoming more and more widely recognized every year. When this principle is universally understood and conformed to there will be little trouble in school government.-C. W. Cole.

The best discipline is not found in schools where the rod is most freely used. It is a rule, with few if any exceptions, that the necessity for the use of the rod diminishes with the increase of skill and power on the part of the teacher. And yet I think it must be admitted that we cannot entirely dispense with the rod. Better secure