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The simplification of the old form of Mach's Apparatus, a piece often made or imported by schools and colleges, consists in three modifications which only slightly impair the efficiency of the apparatus, while they render it cheaper and more convenient.

Instead of the usual rigid supports, the collapsible parallel bars may be mounted on ordinary laboratory standards by means of clamps. The apparatus may then be rolled up and treated like a map or chart, thus saving space,—a most valuable asset in a laboratory.

In order to release the balls in the two representations of reflected or stationary waves, electro-magnets may well replace the cumbrous lever, worked by the foot, seen in the original piece. Better still, a simple gravity release may be used.

Also the same piece that is used to pull off the longitudinal progressive wave may be used to pull off the transverse progressive wave.

The piece was exhibited and put into action,

Among the many methods employed to show wave forms, the speaker preferred some of the stroboscopic methods, as Quincke's.




At the present time a widespread interest has been aroused in mathematics, and some very radical changes in the methods of teaching it have been suggested. As the subject of physics is involved in the proposed changes, it seems but fitting that this conference, the representative body of physics teachers of this state, should undertake to do its part towards finding a solution for the problems thus called forth. I have been asked "to start the ball rolling," and feel I can do so in no better manner than by stating as briefly as possible, first, what seems to be the prevailing sentiment regarding mathematics teaching; second, the general ideas of the proposed changes. My information upon the subject has been derived largely from articles which have appeared during the last two or three years in various mathematical, scientfic, and educational publications. At the best, then, what follows can be regarded only as a summary of the ideas of the writers of

these articles, a majority of whom, it may be worth while noting, are mathematics teachers.

There is a large class of persons, including in their number engineers, physicists, and chemists, who want their mathematical knowledge to be strictly usable. It is the consensus of opinion among these men that it is not usable to a sufficient degree, and they demand that the mathematics taught them be less formal and more practical, and, moreover, there is a growing tendency to the admission that this position is a sound one, that knowledge, so taught that it can be used, must necessarily be more valuable than when taught otherwise.

Physics teachers as a class, moreover, have a grievance arising from the attitude which pupils assume to all things mathematical in physics. While physical ideas are being developed, a class may be all interest, but when the time comes to express those ideas mathematically, the situation changes. The more conscientious prepare to receive the bitter which always accompanies the sweet, the less conscientious ones, while present in body, are plainly absent in mind. This attitude is indicative of the dislike which the pupil has acquired for mathematics during his training in it, and of his belief in its uselessness. It is a severe criticism of the present method of teaching the subject. If physics teachers have anything to ask of mathematics teachers, it is that they endeavor to change this attitude of the pupil to the reception of mathematical ideas. As one can be interested in those things only which one can understand and do, the above situation seems to demand also less formal and more practical mathematics.

Now as to the general ideas of the proposed changes. Great emphasis has been placed on the value of mathematics as a means of mental discipline. So much so possibly that it has often become the end. Prof. Klein of Göttengen has called attention to the fact that another chief value is this: "To make the conviction grow that correct thought on the foundation of correct premises gives mastery over the external world. To do this, attention must be directed to the external world from the beginning." This is the key to the proposed changes. A quotation from a report "On the Teaching of High School Mathematics," read before the Mathematics Section of the Chicago and Cook County High School Teachers' Association, states the idea very plainly. "In present discussions of the possibility of improving the teaching of mathematics, the vital point seems to be that there should be, first, a concrete problem and then its expression in mathematical language, rather than first instruction in the language and then its application to the expression of problems. By problems we mean some real question in the world of senses, not an example from a book. If this is right, the equation has no right for existence till there is first a truth for it to tell. The more the truth told appears to the pupil as worth telling the better." If the pupil then has had experiences which are capable of being expressed mathematically, use them, if not, give him such experiences. As an illustration of the latter case, a spring balance, when the stretches due to various weights are noted, tells the fact that the stretch is always a certain number of times

larger than the weight, say 5, this truth may be briefly expressed by the equation s 5 w. Another balance may give s=8w. Other balances would yield similar results, and a second important truth appears, i. e., all balances have similar equations. This may be told by scw, a general equation, which can be applied to any balance as soon as c is experimentally determined.

Use is to be made of a pupil's intuition and experience. If his common sense tells him that a certain mathematical idea is true, to compel him to demonstrate it before his logical powers are sufficiently developed to make him feel the need of a proof, is to put him often in a state of confusion, as he can see no reason for proving something which is selfevident. On the other hand, if an experimental test of his ideas shows their correctness, he gains confidence in his judgments. So experimental proof may often be substituted for formal demonstrations, and even the conclusions reached by such demonstrations may well be tested experimentally.

If the pupil's mathematical ideas, wherever possible, are thus derived directly from his experiences, will he not regard the subject as a most practical one? Will he not naturally wonder, when he has acquired ideas by observation, if they may not be mathematically expressed? In short, will he not have that attitude to the reception of mathematical ideas which physics teachers would wish him to have?

To furnish these concrete ideas upon which the mathematical ones are to be built, there will have to be a mathematical laboratory. This laboratory, if one judges from the lists of necessary apparatus, might well be mistaken for a physical laboratory. This means that physics is regarded as the subject best suited to give the needed experience, and that mathematics and physics are to be closely correlated or possibly more. It may mean that neither mathematics nor physics, as such, is to be taught, but in their place a single subject which is the result of thoroughly amalgamating the two. That such a scheme could not possibly succeed if applied under present conditions is evident to no one more clearly than to those who are proposing it. Its present practicability is not the question, but rather, would such a training give the pupil a mathematical and physical knowledge which is usable, and would he be filled with a desire to use it? If this question is answered in the affirmative we come to a most practical question of our subject: To what extent could these ideas be put into operation at the present time? In the first place, teachers with a sufficient knowledge of both mathematics and physics for the successful correlation of the two subjects in the manner indicated are very few in number, and progress must of necessity be slow until such teachers are developed. However, if the present situation be accepted as it is, it is possible to make modest attempts in the direction indicated. Such attempts are being tried at various schools, notably at Lincoln, Neb., the Bradley Polytechnic Institute at Peoria, Ill., and at a number of schools in and about Chicago. The general plan of operation seems to be to have in the first year of high school work a course in elementary science, in which the physical laws with which the pupil comes in daily

contact are explained by aid of experiments. Upon the concrete ideas thus obtained, the algebra and geometry taught together are founded as much as possible. This work is continued during the second year with the introduction of elementary ideas of trigonometry. The results are said to be a greater thoroughness and insight into the subject of algebra and geometry. The pupil comes to the subject of physics proper in the third year with definite, usable ideas in mathematics, which include those of positive and negative quantities, ratio both direct and inverse, together with considerable skill in the manipulation of such equations as are ordinarily found in physics.



In induction coil work the ordinary interrupters cease to be of service if the potential difference across the break, when no current is flowing in the primary coil, is high enough to maintain an arc between the contacts, namely, about 40 volts. In the case of a coil energized with current derived from a 500 volt circuit, the drop in potential at the break is many times this voltage if the current-strength is of the requisite magnitude.

The apparatus about to be described was devised to permit the operation of an induction coil on a current accompanied by a potential beyond the range of the ordinary interrupters. The principle involved is the following: A mechanically actuated interrupter of the mercury-in-alcohol type, possessing multiple contacts or plungers, electrically connected in series, and having a large and rapid motion of translation. In detail, the interrupter consists of four glass tubes 5-8 inch in diameter and 8 inches long, placed in a square, zinc-lined box, the dimensions of which are such as to hold the tubes securely in position. The unusual length of the non-conducting liquid is to provide a column of oil or alcohol so deep that its inertia is a factor in resisting and preventing the formation of bubbles of vaporized oil at the moment of breaking the circuit, and which are essential to the existence of the arc. The lower end of each tube is closed by a cork and a layer of plaster of paris. Electrical connection is made with the stratum of mercury in each tube by means of a wire carried down through the central space between the tubes, and terminating in a flat spiral resting on the layer of plaster of paris.

Attached to the lower end of a brass rod, constrained to move vertically by means of suitable guides, is a disc of wood, and from this are supported the four plungers. In order that the plungers may be as rigid as possible, and yet free from any undue weight, they are made of copper tubing, 1 inch in diameter, the lower ends of which are provided with sharply-pointed copper tips. Electrical connection is made with the plungers through flexible

conducting cords. The zinc lining extends above the tubes, and kerosene oil, which is used in preference to alcohol for obvious reasons, fills not only the tubes but the space about them. The motor has a pinion on the armature shaft which meshes with a second gear on the crank shaft. The gearing is on the farther side of the motor and is not shown in the illustration. In order to simplify the mechanical construction, a piece of spring brass was


used to connect the sleeve carried on the crank with the rod supporting the plungers. This rather novel use of spring brass in lieu of the ordinary connecting rod introduces no especial resistance, and answers satisfactorily.

The breaks occur in series: that is, the mercury of tube No. 1, say, is electrically connected to plunger No. 2, and the mercury of this tube is connected to plunger No. 3, and so on. Thus there are produced simultaneously four arcs, and, in order that they may be started at the same instant, it is clearly necessary that the heighth of the mercury in each tube shall be the same, supposing, of course, that the plungers are of equal length. If one of the plungers fails to emerge from the mercury at the same instant the

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