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but only give the semblance of attention. If we seek for the cause or causes of things, I presume we shall find one of the most fruitful sources of evil to lie in this : that in a large majority of instances the amount of time to be spent in study required of pupils just entering school life, is so great, that they are driven in pure self-defense to be listless. It is by no means unfrequent to find that pupils of eight years and even younger are required to study half an hour on a lesson to which it is absolutely impossible for them to give their attention more than ten minutes. From respect to their teacher's wishes, which are evinced in the oft-repeated injunction, “Study your lessons,” or from fear of reprimand, which such injunction sometimes portends, they may preserve the appearance of study beyond the ten minutes, even to keeping the eyes on the book and lips moving, while their minds are wandering upon playthings and schoolmates. The evils of such a course are apparent.
I will not dwell upon them, but turn my attention to answering briefly the question, What shall be done?
1. The teacher must be well acquainted with the work to be done ; must know well how to do it; must be well acquainted with the wants of the class or school and of its individual members ; must be inspired with a love for teaching and filled with the enthusiasm which such love generates.
2. Let the lessons be short. The capacity of the class or of the individual should be so clearly understood and carefully considered that the lesson given may be fully mastered within the limits of proper, healthful exertion. For pupils six to eight years of age, I think the limit may be put at five to ten minutes ; from eigh to ten, fifteen minutes, with three to five minutes for review at the end of half an hour; from ten to twelve, twenty minutes, with a like review ; for older pupils, the time of study may be increased, till one hour shall not be too long, if only followed by the proper
rest. 3. The recitation should not be prolonged beyond the same proper limit of healthful exertion. While studying the pupils should give the whole force of their mental powers to the acquisition of the lesson, so while reciting, they should be thoroughly busy-should each perform every mental operation required of any one. Whatever plan of conducting the recitation is found the best adapted to fix attention, should be followed ; and any plan which does not secure the undivided attention of every pupil should be discarded or modified at once. The teacher's own attention
should be directed to observing whether the pupils are thoroughly interested and absorbed in the work to be done, and the moment the interest abates the recitation should be closed. Better no recitation at all, than one continued while the pupils are thinking of something else or nothing at all-dreaming, as so many do. Many complain of want of time to conduct the recitation. The trouble is not want of time, but want of proper employment of time. More time is wasted than used, if my observations may form a proper basis of judgment.
4. The teacher should never allow any portion of a recitation to pass without giving to it his own personal and undivided attention. The habit of assisting individual pupils to do "sums," or to find localities on the map, while others are reciting, can not be too pointedly condemned. If the recitation is not worthy of the teacher's attention, how can he expect to command the attention of the pupils!
5. While the recitation is in progress, the teacher may ask frequent questions for the class to answer in concert. Let the questions be such as all can answer, and let the teacher see that all do answer.
Many more like suggestions might be added, but I forbear. Enough has already been said to set the live teacher to thinking; and more might only serve to harden the plodder in his dullness.
0. S. c.
GENERAL PRINCIPLES IN MATHEMATICS.
BY A. HERRMANN.
As reason constitutes the supreme and ruling principle in human affairs, the value of correct reasoning both for individual and general welfare can not be overrated. Mental discipline, therefore, forms one of the principal ends, for which the institutions of public instruction in our country are established. Its importance is at least equal to the acquisition of a certain amount of objective knowledge, whose utility no sensible man will deny. This holds true, not only in respect to institutions of learning of the highest order, such as colleges and universities, but also in regard to all our common, intermediate, and high schools.
But one of the chief means of forming habits of strict, sound, and correct reasoning is furnished by mathematical instruction in
its different branches and grades. Especially in our high schools it may be used, in some measure, as a substitute for properly philosophical instruction, and, at the same time, as an introduction to it,—as a kind of propædeutics for the truly scientific exposition and study of logic, metaphysics, and mental philosophy, which, of course, must be reserved for the university course. Yet the full realization of those benefits supposes that the teachers themselves possess a sufficient insight not only into the general principles which underlie and govern all mathematical development, as far as the same bave become known by the progress of science, but also into their relations to the nature of mental activity and the laws upon which the soundness and correctness of human reasoning in general depend. These principles are the very life of mathematics, and they alone are able to elevate the instructor above the level of a mere bungler, making him thorough master of what he professes to teach. The truth of this is, at least, partially admitted by acknowledging that the Pythagorean Theorem and the propositions concerning the similarity of triangles are alone sufficient to prove all theorems of common, plane geometry, and that even the former can be deduced from the latter.
No doubt the excellent or even prominent teachers of mathematics who adorn the various high schools of the country, are well aware of the truth and importance of the previous remarks. Still it is no wonder, if, in the practical application of them, we have not yet fully arrived at that degree of perfection which is both desirable and attainable in the present condition of scientific culture. The only book written in this country, which has come to my knowledge as having for its end a full, systematic development of the philosophical foundation of mathematics, is Davies' Logic of Mathematics, -excellent in many respects, yet unsatisfactory in others; and it seems not so extensively used and studied by teachers as it deserves to be.
For the purpose of illustration, I will mention here a couple of facts taken from my personal experience, merely for the common good, and with the hope that no personal offense will be given by this communication :
1. It is a well known axiom used in mathematics, that things which are equal to the same thing are equal to each other. An obvious deduction from this principle is, that quantities which are equivalent to the same quantity are equivalent in regard to themselves: which conclusion is expressly or implicitly acknowl
edged to be correct in every work on geometry. But it follows from the same axiom with equal necessity, that things which are similar (in a geometrical sense) to the same thing are similar to each other. This can be proven strictly after the method of Euclid. For suppose two triangles ABC and DEF similar to a third one, GHI. Then the angle B=E=H; and the side AB: CB=DE : FE=GH: IH. On the ground of logical principles the correctness of that conclusion will not be doubted by any one who has read Sir W. Hamilton's Lectures on Logic. Of course, it is legitimate only in mathematical science, in which the term “similar" possesses a strictly definite meaning. It would not be correct to conclude that two persons must-always or in every case—be similar to each other, because they are similar to some third person ; for they may, nevertheless, be quite dissimilar if compared among themselves. In other words, this conclusion is wrong, because the middle term of the syllogism is not distributed. But I could not help, some time ago, to be a little surprised when I found that one of our excellent teachers of mathematics hesitated to acknowledge the correctness of the above conclusion in regard to geometrical similarity.
By this I do not intend to say, that it is advisable to use general principles more than it is absolutely necessary, for the instruction of beginners in the study of geometry. But I dare say, that it will prove a great benefit to direct the attention of the more advanced scholars to the existence of such principles in cases where their application is apparent, and to show the use of the same in simplifying the science and pointing to its connection with the philosophical foundation of human knowledge. This method will occasionally offer an opportunity to open views in the very depths of knowledge, which are apt, more than any thing else, to make of the student a decided, enthusiastic votary of scientific research by inspiring his mind with the love and desire of deeper insight.
2. What has been said concerning the importance of knowing the general principles which form the basis of mathematics, both for a thorough understanding of the latter and the promotion of mental discipline, holds also true, with some modifications, in regard to other branches of knowledge, and especially physical science; and as this latter is intimately connected with mathematics, a thorough knowledge and correct application of the principles contained in it is often necessary for a correct solution of mathematical problems. As an illustration let us take the
well-known problem : What must be the inclination of the roof of a building that the water may run off in the least possible time?
The usual way to solve this problem is as follows: Let a designate the base of the roof, and x its altitude: then v a2 + x2 will be its length, and
a the time required for the water to fall (perpendicularly) through the height of the roof. But the time down an inclined plane (t) is to the time through its perpendicular, as the length of the plane to its height.
Vaz + x2 Therefore,
ga Because t shall be a minimum, the differential coëfficient of the second member is 0; and g denoting the force of gravity (16.1 feet the first second) Vų is a constant factor, which can be omitted in the process of differentiation.
Therefore, 2:02 a2 X2=X2- a2 0; or x =a?; consequently, x= a; i.e. the base of the roof is equal to its height, or the triangle is both rightangled and isosceles, the angle of inclination being 45°. When the roof is double or inclined on both sides, the angle of the ridge will be 90° or a right angle.
I remember that this problem was proposed for solution, some years ago, in the State Journal of Education of one of the Western States, the mathematical department of which journal was under the supervision of the Professor of Mathematics at the State University. The teacher who furnished the solution (about in the same way as it is given here), remarked that the influence of friction had not been taken into account; and this induced the editing Professor to put the question: Will not some one solve the problem, taking into account also the friction ? Now, it seems to me, these remarks show that both the editor and the author or copier of the solution had not fully understood its nature and bearing. For, although it must be admitted that in the solution as it is given before, the influence of friction is not mentioned, yet a little reflection will convince every one who is acquainted with the physical law referred to, that friction, whatever may be