HINTS ON GEOMETRY TEACHING.

By G. C. SHUTTS.

State Normal School, Whitewater, Wis.

It goes without saying that in this country and in this day of geometry teaching that our aim in teaching geometry is to encourage the pupil to do the maximum of original demonstration and construction and to use any legitimate authority without regard to an arbitrary standard or text; that the standard of success for the pupil in a given demonstration is: "Have I given authority that we have already established or agreed upon for each claim that I have made from premise to conclusion?" When the pupil has acquired this power to select the proper authority and make accurate application of it, the function of the teacher is accomplished, for the pupil is self-directing and can master the subject by himself. He has in part acquired and is acquiring the mental power for which the subject is generally taught. All that follows in this discussion is based upon the above assumptions.

Probably the most difficult thing to accomplish in teaching geometry is to get the pupil in the habit of studying carefully to determine what is to be proved and what is given him in the theorem as a basis for the proof. In the assignment of the lessons in the early part of the course, it is well to have the pupils again and again select from the theorem this data. If the thing to be proved is indicated by a defined term, as parallelism of two lines or of two planes, a parallelogram, etc., the pupil should state what is to be proved in terms of the definition.

It is a common remark that the mathematics of the secondary school should be considered as a tool to be used in other subjects or in life. This, like many another, is but a partial truth. Each of the mathematical subjects becomes to the pupil who must terminate his school life with its completion, if it has been truly a thought study, a vital possession, a source of genuine pleasure, a means of acquiring conscious power. But these subjects to a certain extent are tools, and as tools they should be made as efficient as possible; and it is probably true, that the more it is appreciated that a given subject is a tool, and the more efficient it becomes as a means to a given end, the more truly will its other values be appreciated. Hence to get the fullest value in this line from geometry, each axiom, definition, theorem, etc.,

should, when once acquired, be regarded as an instrument with which to do further work; hence the more thoroughly it is mastered the more valuable it becomes. The pupil should be tested as to whether he can recognize the theorem in different relations; he should know the essentials from nonessentials; should be given. opportunity to recognize it with many as well as with few nonessentials. A reason why pupils are so slow to use established truths in original work, is that they have seen their application in but one form usually lettered in the same way, or they have not brought it to bear in any way upon matters outside of the school

room.

For instance, the isosceles triangle may never be expressed upon the blackboard with the vertex down or at the right or left of the base; it may never occur to the pupil that the corner of a sheet of paper, or hundreds of other things about him, is a representation of his notion of a right angle. If he be given a carpenter's square, or other right angle, and be asked to test its accuracy by the truth he has acquired, the value of the truth becomes at once more fully appreciated as well as the truth itself more fully known. When he has learned that two triangles are equal if two sides and the included angle of one are equal to two sides and the included angle of the other, respectively, if he be asked to apply it in finding the distance between two points that cannot be measured on the straight line between them, he feels that geometry is a part of life.

Let the value of a theorem be shown by using it in the demonstration or solution of exercises. When an exercise is proposed, let the pupil rehearse all the theorems that may refer to the topic in hand, and from them select one or more that are suitable for use. It may be profitable to make several solutions. For instance after several tests for a parallelogram have been proved let each one be used in establishing a new test that may be called for. Or when a third or fourth proportional is to be constructed, let the pupil rehearse all the theorems in which four terms are in proportion; let him study the figures involved and effect a construction by means of two or more of them. For instance, construct a fourth proportional to three lines by two chords intersecting in a circle; by constructing two similar triangles; by two parallel lines intersected by lines drawn through a common point; etc. A third proportional con be constructed by each fourth proportional proposition and also by each mean proportional

proposition; a mean proportional by each of the latter. By a large use of theorems as tools in construction, interest in the subject as well as the efficiency of the subject as a tool is enhanced. In effecting constructions, as elsewhere, the pupils need to be taught how to study: One way to accomplish this, after agreeing upon a previous theorem for use as a tool, is to represent this theorem by a figure and study its properties, to see how it may be used in making the construction. For instance, suppose we wish to construct a third proportional, x, to lines a and b, the latter being the mean proportional, using the theorem that a perpendicular from the vertex of the right angle of a right triangle to the hypotenuse is a means proportional between the segment of the hypotenuse. When the general figure is drawn, ask such questions as: Where in the construction must b be found? Where is a? When will x be found? Which lines, a and b being given, can first be drawn? How are a and b related in the figure? Then by what authority can they be drawn? Which line can next be drawn? How is it related to a and b? Then how can it be drawn? Which line next? Authority for its construction? etc.

A grave difficulty in solid geometry in attempting to do original thinking, is in the use of authority from plane geometry. The pupils are apt to take the authority literally and forget that a common fact is assumed in all statements in plane geometry but not stated, viz.: that all the lines are in the same plane. Sometimes the authority to be used is of such a nature that the lines must lie in one plane, as the lines of a triangle or a line perpendicular to a line; in such cases no mistake can be made. But the lines of an authority may be in different planes if the above assumption is not insisted upon. For instance, the authority, "at a given point in a line one and but one perpendicular can be erected." If the pupil wishes to apply this theorem in an indirect demonstration, he is apt to forget that the two perpendicular lines and the assumed line must all be in the same plane, for any number of lines perpendicular to the same line at the same point are possible in solid geometry. Again, suppose the pupil wishes to prove that the intersections of a plane with two parallel planes are parallel lines. First, "What is to be proved?" "That the lines are parallel." "But when are lines parallel?" The pupil thinks of the definition, and says, "When the lines. can never meet." This form of statement, true for plane geom

etry, is not true for solid geometry. Hence the pupil in his demonstration must call attention to the fact that the lines in question are in the same plane, as well as that they can never meet, if he would do thoughtful and accurate work.

It is advisable to group in review all propositions having a common element; as, propositions on parallel lines and transversals, lines perpendicular to a line, tests for determining when a quadrilateral is a parellelogram, tests for determining that two triangles are equal, etc.

If the teacher is also the instructor in physics or engineering, etc., he can help the pupil to anticipate this subject by getting the geometric tools ready for use in it. But it is not wise to assume that this is the main function of geometry. A persistent application of parts of the subject in the development of other parts, with a wise selection of subject matter, together with a determined and courageous effort to limit the amount of matter to the time of the course and the ability of the pupil, will probably make the subject abundantly vindicate its demand for a place in the curriculum of our secondary schools.

THE TRAINING OF THE INDUSTRIAL CHEMIST.
By W. D. RICHARDSON,

Chief Chemist, Swift & Company, Chicago.

My point of view is that of one who wishes to see the graduate in chemistry enter into his life's work well enough equipped to insure him a fair degree of success, without to great delay. Most persons recover but slowly from their first failure. Further I have in mind only the man or woman who can devote but four years to the work of preparation. I have nothing to say of the one who is free to spend seven or twelve years at a university. Of the difficult problems which confront our educators I am not at all qualified to speak, and I trust that nothing I have to say will offer insuperable pedagogical difficulties. I feel indeed somewhat apolegetic, that I should have the temerity to speak of education before educators, since I am in nowise a teacher. But I have come to believe that the schools and colleges and universities are not the only educational institutions. Are not the large industries of the country also educating large numbers of people from day to day? The methods are different, but the results have points of similarity. It is only because I feel that I too am

on the faculty of an educational institution that I presume to address you at all.

My first thought was to name my talk this afternoon "The Trade of Chemistry vs. the Trade of Bricklaying," for I could. then tell you directly that the latter trade is as honorable, no harder and more remunerative that the former. The apprenticeship in the bicklayers trade lasts about four years. In the trade of chemistry in the work's laboratory about as long. The bricklayer earns from $25.00 to $35.00 per week, the journeyman chemist from $15.00 to $25.00.

But let me go back a step. It may not have occurred to you that there is a trade of chemistry as well as a profession of that name. I designate as a journeyman chemist one who as a young man spends a certain length of time in the laboratory of a manufacturing establishment, among trained chemists learning how to make routine analyses. In all large laboratories there are a number of such men working. They come from various ranks, as they would to learn any trade. The reason is that the monotony of routine work soon wearies and discourages a properly trained graduate and incapacitates him for the work he is better qualified to do. The making of thirty or forty nitrogen determinations daily, after the art is once well learned, does not require a great knowledge of chemistry, nor does it tend to develop the broadest kind of chemist. At the same time this sort of work requires all the skill and watchfulness the average man can muster. In every larie laboratory there is much work of this sort and it, as well as other work, must be handled.

There is nothing more illuminating to the college graduate than his entering a work's laboratory, there coming in contact with boys and young men who without any college training are making chemical analpses in the most approved way. He soon finds that it is only by the greatest endeavor and hard application on his part that he can compete in any degree with these employees. He discovers too to his chagrin that these men are naturally about as capable, about as clever and considerably more inured to continuous hard work than himself. His observations will at first discourage him in a large measure, but at the same time will lead him to inquire as to what particular merit his college training has conferred upon him, and in his answer to this question and his resultant action will lie in large degree the reason for his future success or failure. Moreover at this point the