THE TEACHING OF ELEMENTARY MECHANICS.

In considering the question of the teaching of Elementary Mechanics in this country, a brief glance must be taken at the history of its development. Engineering and Physics, art and science, practice and theory, have advanced on independent lines, and have only recently amalgamated, and the growing importance of Mechanics as a school subject is due to this amalgamation. Shipbuilding, to take one instance among many, was formerly an art calling for much expert knowledge and applied experience both from the designer and the workman. It has become a science in which experiment and theory are continually working in double harness, and the subdivision of labour, along with the standardization of materials, has taken much of the responsibility from the workman, and laid it on the naval architect. The same thing is true in bridge building, carriage building, and all kinds of mechanical engineering. In addition the applications of electricity have produced a new company of electrical engineers. It would seem that, roughly, one boy in five wants to be an engineer, and it is recognised that a severe mathematical training is wanted for all such boys. The training is provided by Engineering Colleges and Technical Institutions; but many of these require their students to pass an entrance examination in which Elementary Mechanics is a subject. We have therefore a large number of boys who require a course of Mechanics at school with direct application to the requirements of their future profession. There exists also the comparatively small class of boys who are working for mathematical scholarships, and a somewhat larger class of Army Candidates. For these Mechanics is merely part of their general education, a branch of Applied Mathematics. But the movement of approach between practice and theory has manifested itself on both sides, and while the practical men have become more mathematical, the mathematicians on their side have in recent years made a determined effort to make the study of their subject more practical.

The practical teaching of Mechanics in schools is almost entirely a modern growth; Professor Worthington's Physical Laboratory at Clifton was started not more than 30 years ago. Now a large number of schools possess physics laboratories where some mechanical work is done; but even now there is very little connection between it and the mathematical teaching. To quote from a report, published in 1909, made by a Joint Committee of

the Mathematical Association and the Association of Public School Science Masters" :

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"In no subject is the want of co-operation between the mathematical and the science masters so apparent as in Mechanics. In the large majority of schools Mechanics is taught by the mathematical master as a part of the regular mathematical work; no experiments are made, and in very few cases are practical difficulties mentioned; this instruction is confined to the senior boys, with the result that the science master finds the boys who come to him are ignorant of such conceptions as the principle of moments, or the parallelogram of forces, and unable to distinguish between force,' 'work,' and 'power,' and is obliged to put most of his younger boys through a short and necessarily inadequate course in Practical Mechanics, in order that they may be able to follow his lectures. There is no connection between the lessons given by the two masters, and it too frequently happens that the better mathematicians leave school without having done any practical work in Applied Mathematics, whilst many promising science boys are deprived of the valuable training afforded by the mathematical treatment of this subject.

“The requirements of the Army Entrance Examination have caused a certain amount of practical work to be done during the past few years, but even now rather less than half the public schools (which are those chiefly affected by the regulations of the Civil Service Commissioners) report that practical work forins a part of the regular teaching in Mechanics."

The main defects in organisation appear to be briefly these :

(1) The very short course of Practical Statics which a teacher of Physics has to give as a preliminary to the study of Magnetism and Electricity has no connection with the mathematical teaching;

(2) Boys working for mathematical scholarships not only frequently miss even this short course (a comparatively trifling matter), but they also have few opportunities of acquiring a practical acquaintance with more advanced mechanical ideas.

The matter of organisation will probably right itself in time, more especially as mathematical teachers are becoming more and more accustomed to laboratory work. The present paper is concerned with the actual details of teaching rather

The Correlation of Mathematical and Science Teaching. Report of a Joint Committee of the Mathematical Association and the Association of Public School Science Masters. (London, G. Bell and Sons, 1909.)

than with the question whether Mechanics is to be taught as part of Mathematics or of Physics.*

It is assumed that whether practical work and theory are taught by the same person or by different persons, the work will all form part of the same scheme, and for the purposes of this paper it is taken that the laboratory work is under the control of the mathematicians, as in the case of Professor Henrici's Mechanics Laboratory at the Central Technical College. This institution, as a place of training for engineers, and as a college and not a school, comes outside the scope of this paper. But as the inferior limit of age is 16, and ast the first year's course for all students includes Elementary Mechanics, it may be as well to mention how that course proceeds. Before any lectures on Statics are delivered, i.e., during the whole of their first term, students spend three hours during one afternoon in each week at practical work in the Mechanics laboratory. Then, when the time for lectures on the subject comes round, they are all familar with the things signified by the words employed by the lecturer. The same plan is adopted in the teaching of Kinetics, which follows that of Statics. Naturally there must be great variety in the ability and previous knowledge of the students, many of whom are as old as 19, and all of whom have passed the entrance examination in Mechanics; and their attitudes towards the laboratory work are as various as their attainments; but they all go through the course, and it produces a marked effect even on those whose book knowledge is already considerable. It should be noted that the laboratory in question is mathematical only, under the direct control of the Professor of Mathematics, and quite separate from the engineering laboratories to which students proceed later in their courses.

Before we consider the problem presented to schools of the ordinary public school type, there remain two other institutions providing for a special class, from which much may be learnt. The Naval Colleges at Osborne and Dartmouth carry on the training of naval officers between the ages of 12 and 161.

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Before the end of the first two years, spent at Osborne, cadets have begun Trigonometry and have been through a course of Practical Statics, devised with the purpose of giving these young boys "a certain number of ideas on the subject of Mechanics, and breaking the ground for a more complete "treatment of the subject at Dartmouth." The work is almost entirely statical, though notions of potential and kinetic energy are treated in general terms and without formulæ. Special features of this elementary course are the detailed treatment of pressures and tensions by means of spring balance experiments, and the measurement of work with particular reference to the efficiency of tackles, screwjacks, Weston's pulley, &c.

A pious hope may here be expressed that before many years are past the two subjects will be taught by the same man.

This simple practical course is now followed at Osborne by a more detailed mathematical treatment, by which the elementary notions of the subject are "rubbed in.'

At Dartmouth, where the second two-year period is passed, a more complete course of Statics is followed by Kinetics treated inductively and experimentally throughout. Newton's laws of motion are not assumed, and the rest deduced, as in the ordinary mathematical course; but at each step experimental work is done as a preliminary. The use of formulæ is not encouraged, except in the case of the more able mathematicians; but in all cases an effort is made to ensure that the physical facts lying at the base of the calculations are grasped by the students who make those calculations.

In the case of the ordinary public or secondary school, it may be said that, apart from the short course already referred to, Mechanics does not form part of the general education of all boys. This may be regarded as unfortunate, inasmuch as a reasoned Mechanics is one of the intellectual possessions by which the modern world is distinguished from the mediæval; but the cause is not far to seek. The average boy does not carry his study of Mathematics to the point at which Mechanics is usually introduced, and unless universities and other examining bodies begin to regard it as a necessary part of a general education, it may safely be prophesied that the average boy will not reach that point. It would be possible for teachers of Mathematics to advance the stage of beginning Mechanics by shortening the course of Algebra and introducing the elements of Trigonometry along with quadratic surds and ratio, the laboratory course of Mechanics being carried on simultaneously. The whole question is, for schoolmasters, complicated by the requirements of examinations, and as long as Elementary Algebra is demanded as a subject by itself, divorced from its applications, much time must be occupied in a process similar to that of teaching swimming in a dry gymnasium. schedules of examining bodies fall outside our discussion, the scope of which may be outlined by the following headings: (1) The age of beginning the subject; (2) its relation to Trigonometry; (3) the question of taking Statics or Kinetics first; (4) the amount of practical work desirable from the mathematical point of view; (5) the need of a laboratory. It is at once obvious that under existing conditions the different classes of boys already referred to must be considered separately in discussing these points. But in view of future possibilities it seems more fitting to range the boys in two classes, not from the examination standpoint but from that of mental capacity. We will not call them (a) boys who are going to an engineering college and (b) boys who are working for a mathematical scholarship, although this is what may actually be the case. We will rather call them (a) boys of a practical turn of mind without conspicuous mathematical ability, and (b) boys who

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find Mathematics easy. The first of these classes is very much larger than the second, and it is with it that the first head of our discussion is particularly concerned.

The age for beginning Mechanics.—We assume that a course of elementary physical measurements has been worked through by all the boys whom we are now considering, and that length, area, volume, mass, density, and Archimedes' principle have been dealt with in the way which has become almost universal in schools of the type in question. The practical work should have served the purpose of making mathematical work more of a reality, and if the course adopted is similar to that outlined in the Report of the Joint Committee already referred to, recommendations 1-6, it will have done this. Recommendation 7 runs as follows:

"The term 'measurements' as used above should not be understood as including any form of Mechanics; e.g., even simple experiments upon the principle of moments or the parallelogram of forces should be postponed until the pupils have gained some acquaintance with the elements of Trigonometry."

Up to this point the "measurements course has been intended for all boys, whatever their mathematical ability. A stage is reached where an Elementary Science course for all boys should begin. The Elementary Physics course usually includes Heat, Light, and Magnetism and Electricity. Should it include Mechanics?

The Physics course now takes two different aspects. The science teacher must perforce regard it as of primary importance, and Mathematics as its handmaid. The mathematician's standpoint reverses these positions. For the present let it be granted that we are still using our practical work to help on the Mathematics. The boys have been helped to recognize the advantage of algebraic formulæ for mensuration purposes, they have mastered the decimal point and approximate methods. Before they get much further they must learn indices and logarithms. Elementary Practical Mechanics is not of much use here. A subject like Heat presents more practical applications of the use of logarithms, and Mechanics may very well wait until the beginnings of Trigonometry are approached. After Heat the Physics course very likely takes Magnetism and Electricity. It is here that the science teacher finds the need for a little Trigonometry and Mechanics. Probably the boys have not done any yet, and they have to be put through "a short and necessarily inadequate course in Practical Mechanics."

Is this a bad thing? Ought the mathematical master to hurry the boys on, so that he may provide a groundwork for the teaching of magnetism? To the writer it seems that no harm will be done by this short and inadequate course, provided that a sufficient understanding exists between the teachers of Mathematics and Physics to make the Practical Mechanics