ORIGINAL DEMONSTRATIONS IN GEOMETRY

I. PURPOSE, NATURE AND METHOD OF PRESENTATION

FLETCHER DURELL, TEACHER OF MATHEMATICS, JOHN C. GREEN SCHOOL,
LAWRENCEVILLE, N. J.

It has often been said that working original exercises in geometry constitutes one of the best means at our command of cultivating in pupils the power of clear, logical thinking. The question now raised is whether there is not some way of giving this educational discipline a deeper significance, and making its fruits available to practically all pupils.

It is also a common remark that in geometry the pupil has before him a set of tools, viz., the triangle, line, point, etc., which he must learn to use; and that in working original exercises, he must realize what tools he has before him and learn to select that tool or combination of tools best fitted for the work in hand. Perhaps something can be gained by carrying out this idea farther than is customary, by studying the utilities and properties of each geometric implement, getting the most practical order in which to master these implements, and the most advantageous way of taking up each one.

The question at once arises, which tool occupies the primary place? Which is used most, is grasped most readily by the pupil, and logically occupies the most central position. An examination of the 165 text theorems in a standard plane geometry shows that the triangle is used ninety-five times as an aid in demonstrations, while the next most common tool, parallel lines, is employed only forty times. Every attentive teacher must also have noticed that after the pupil has learned the use of the triangle he prefers to use this tool wherever possible. The triangle seems to mediate between the simpler elements, the point and line, on the one side, and the more complex, the polygon, the circle, etc., on the other. In like manner, far more applications are made of equal or congruent triangles than of the triangle in any other form. It seems natural then in teaching originals in geometry, to let the pupil master the use of congruent triangles first. Let this be done by arranging the propositions of the text so as to use the triangle as early and freely as possible, and also by arranging original exercises in groups, putting first a group of exercises in proving triangles equal. If the pupil is weak and cannot use equal triangles in the seven or eight forms in which they appear, limit the subject matter at first so that he has to choose between only two or three of the tool forms of equal triangles.

After the pupil has acquired the power of proving triangles equal, let him take up a group of exercises in which he utilizes this power in proving a pair of lines equal by means of equal triangles. This constitutes both a review and an advance. Next give him a group of exercises in which he is to prove a pair of angles equals by use of equal triangles, thus putting the tool whose use has just been mastered to a new application.

After mastering the use of the first geometric instruments thoroly in this threefold way, the pupil can proceed more rapidly. The use of parallel lines comes next, both naturally to the pupil, and logically in relation to the subject, since two parallel lines and a transversal form but a special case of the triangle. After this the pupil can be made to feel that the locus for some purposes is a more efficient tool than the triangle. Thus ask him to prove that the diagonals of a rhombus are perpendicular to each other. He will naturally do so by the use of triangles. Then bring him to realize that the desired result can be attained with half the labor, or less, by the use of the locus, and he will give this new tool a cordial welcome. At this point it is usually well to introduce from algebra the use of a symbol for an unknown quantity and of the equation, and have the pupil realize that these tools tho mastered in connection with another subject are useful in geometry also.

So far, no use of auxiliary lines will have been made. But now that we take up their use, a group of exercises will be given each of which calls for one or more such lines, and the pupil will be kept at work at this tool till it is mastered thoroly. Similarly the mastery of various other geometric tools, as inscribed angles, similar triangles, analysis, symmetry,

etc., may be acquired. Occasionally groups of miscellaneous exercises are to be introduced compelling pupils to use in combination all the instruments hitherto mastered.

In some cases it is worth while to bring out the toolage properties of a single theorem. Thus from the fact that two triangles are congruent if the three sides of one triangle are equal respectively to the three sides of the other, it follows that a triangular frame of iron or steel is rigid even tho hinged at the vertices. This is the unit of rigidity in girders, bridges, frameworks of all kinds, and is of vast practical importance. The value and use of this unit tool can be brought out by problems like the following: Will a quadrilateral frame hinged at its vertices be rigid? Will it be rigid if two points, one in each of two adjacent sides, be joined by a bar, and if so, why?

Sometimes tool relations (often reciprocal) can be mastered in groups. Triangles are useful in proving the equality of lines and angles. Lines and angles are useful in proving the equality of triangles. Triangles are useful in proving lines proportional, and vice Lines and points determine each other reciprocally; so of lines and planes and points and planes. Some classes are much interested in making lists of the serviceable properties of each geometric tool or combination of tools and making comparisons of these.

versa.

Beside original exercises in direct demonstrations and constructions, of which alone we have been speaking thus far, there are other original or semi-original exercises to which the same method of treatment applies, the development of power to handle successive tools in these different lines being contributory to one common end.

Thus geometric drawing and observational geometry, even though treated in a very limited way, may be made to illustrate the tool functions of the point and line. Geometric drawing constantly illustrates the fact that two points determine an entire straight line, three points determine three lines, and the greater the number of points, the greater the possible efficiency of each point.

So the drawing of lines and angles may be made to illustrate the utility meaning of the axioms, and show that the axioms are to be regarded not so much as fundamental equivalences, as fundamental utilities.

Similarly, the economic functions of straight lines may be brought out from another point of view by showing their utility in the mensuration of areas of volumes. To make two or three linear measurements of a tank involves far less labor than to dip oil out of the tank and count the number of gallons dipped out. Hence questions like the following arise. The measurement of how many and what lines is adequate for the mensuration of the area of each plane figure, and for that of the surface and volume of each solid figure? Under what circumstances is it advantageous to use one linear measurement, or one set of linear measurements, rather than another? Thus, given that the mensuration of the area of the circle depends on the measurement of one of three lines, the radius, diameter, or circumference, which of these three lines is it most convenient to measure in determining the area of the cross section of a pipe whose ends are inaccessible (as in the hot-air pipe of a furnace)? In so simple a case even it is with a distinct thrill of appreciation and sense of new power that the pupil realizes that it is best to obtain the area by measuring the circumference.

Other concrete applications give a new and deeper grasp of the toolage properties of lines in relation to other objects. For example, compare transportation by rail with that by water. A railroad train approximates the line in form and needs a comparatively large and scattered crew and much apparatus to manage it. A steamboat has three dimensions and is, so to speak, concentrated near a point, whence it can be managed by a small crew. This difference is the chief source of superior economy in transportation by water over that by rail, and it is a principle which has important applications in many other fields.

In like manner the study of numerical exercises in geometry may be facilitated and made more significant by converting it into the mastery of a succession of tools. We may separate the power to handle the formulas from the power to carry on difficult numerical computations and learn each separately. First have groups of exercises involving the

numerical relations of various geometric objects in all their combinations and permutations, but in terms of small and exactly related numbers. Afterward have exercises adapted to develop the power of carrying on a long and complicated numerical computation. In this connection, but to a certain extent apart from it, groups of exercises adapted to give mastery of the most advantageous order of operations, of cancellation and other useful devices in computations, are to be worked.

A word may now be said about the application of the above method of studying original exercises to actual classroom work, and then something as to results obtained. In Book I, as soon as the pupil by study of the text propositions has become acquainted with hat is meant by a demonstration and with the use of the triangle, the class may be put to work exclusively on easy originals in proving triangles equal, and in proving lines and angles equal by means of equal triangles. From four to six exercises may be assigned at a lesson, and three or four more solved as sightwork, either at the blackboard by the entire class under the supervision of the teacher, or on paper as testwork. Thus after the class is under way practically an entire group is covered at a lesson. After the class has learned to use congruent triangles in original exercises, we return to the study of text theorems, but in connection with each daily lesson thereafter, more or less original work is done in various ways, as by assignment as part of the regular lesson, or as sightwork. At the end of Book I, more daily work in originals alone is taken up, and similarly thruout the subject.

By this means, every pupil who can learn any geometry learns to do originals, usually takes more interest in the latter than in textwork and succeeds better at it. In my own work, no pupil is passed in plane or solid geometry, or is certified as prepared to take a college entrance examination in these subjects, till he has passed a test in originals alone, as well as one in numerical exercises, and one in text propositions. Besides these direct results, the method of treating the subject outlined above gives other results which are of interest. It seems to have a stimulating and vitalizing effect on the whole subject of geometry. Many proofs which before seemed forced and arbitrary now become direct and natural. Every proposition comes to have meaning and value. The method seems to have an invigorating and vitalizing effect on the study of branches of mathematics taken up subsequently, as on trigonometry and higher algebra.

Some help is given toward the solution of the problem of the relation of intuition to logic in the study of geometry. For my part, I do not see how we can dispense with some intuition, that is, preliminary visualizing in geometry, or in any subject for that matter. For instance, the idea even of a straight line or a circle, is a highly complex one when considered apart from the intuition which we have of such an object. How do we know by pure logic that it is possible for a line to exist, all points of which are equidistant from a given point? But if the pupil makes a start somewhere in geometry, as near first principles as possible, for the sake of clearly realized advantages, later he will be ready to deepen and sharpen his logic for the sake of other plain advantages.

The history of geometry comes to have new interest. It becomes the study of a progressive series of new and improving tools. As a result of such study the pupil sometimes imbibes some of the dialectic of history and is seized with the ambition to invent new tools for himself. The problem of combining individual instruction with class instruction is also partly solved. If originals be grouped as indicated, it is possible to assign as a lesson, say three easier originals for the whole class, and two more difficult ones to be worked by the stronger pupils only. The whole class advances together, but its more gifted members gain a wider and deeper grasp of the subject than the rest of the class. The method is thus an important economy to the teacher who has large classes to conduct.

In like manner, it prevents the somewhat narrowing effect which the study of geometry is apt to have. If a pupil forms the habit of acquiring the mastery of a succession of abstract tools whose values are consciously realized, he should be ready to welcome any new and better tool wherever found. Hence we arrive at a correlation of studies which is not superficial and concrete, and so full of varied detail that someone has termed it not correlation,

but conglomeration; but at a unification by means of underlying utility principles. For instance, if the pupil in studying geometry acquires the habit of selecting the best linear measurements on which to make the mensuration of the circle, cylinder, etc., depend, when he comes to physics he will be ready to apply the same idea to the measurement of physical magnitudes. If he learns to realize the advantage of using auxiliary lines in the vivid field of geometry, he should be ready to use auxiliary objects freely in physics, chemistry, engineering, in ethics, sociology, and in fact find in such use a principle having a hundred applications for him every day.

Before I close, allow me to say a word on one other aspect of the subject in hand. For the past generation, educational theory and practice have been mainly inspired and controlled by the doctrine of evolution. The evolutionary philosophy of education has taken up and dominated other ideas, as those of Herbart and Froebel. We owe to this philosophy of education much of theoretic value and practical worth. But as a whole its tendency is brutal and materialistic. In the nature of things it is but a preliminary step. For we no sooner announce the doctrine of the survival of the fittest, than the question arises, what is fitness? What is value? What is worth? Hence a new philosophy, termed pragmatism is springing up, which is trying to answer these questions and give a broader and higher view of things, which shall include the doctrine of evolution as a mere detail. Perhaps it may be possible to analyze fitness into the elements which compose it, and then devise economical and efficient ways of mastering these elements. Perhaps also after we have made this analysis we may be able to arrive at some more fundamental and inclusive category than fitness. Something worked out along this line must, in the end, profoundly influence educational theory and practice.

As a step in the new direction, let us make a distinction between concrete and abstract utilities. The use of similar triangles to determine the height of a steeple is an example of concrete utility, or of putting a geometric tool to a concrete use. The use of similar triangles to prove a set of lines proportional, or a pair of polygons similar, is an example of abstract utility. Now if the pupil's mind be fed on the concrete applications of mathematics alone or preferentially, a low grade and materialistic appetite is generated in him. His outlook and tastes are apt to be narrowed and he will be satisfied with nothing that does not have immediate material application. If, on the contrary, he gets an insight into abstract utilities, he finds that these are comprehensive and enlarging. They include practical applications as details, and suggest other particular concrete, as well as other more general abstract utilities.

Now the method of teaching originals presented above may be termed the method of abstract utilities, inasmuch as it puts abstract uses in the first place. Concrete utilities are brought in occasionally to sharpen and correct conceptions, and make interest more vivid, but abstract utilities are omnipresent and controlling, and fashion the development of the subject. When thus realized, the method which we here advocate is seen to have new and wider values than those mentioned hitherto. The ends which we aim at in the study of geometry are, first, practical results; second, general culture. The ancient Egyptians in their use of geometry in land surveying, temple building, and barn measuring, sought only the former. The Greeks aimed only at the latter; we desire both. In attaining the mastery of abstract utilities we, in a measure, attain both. The idea of abstract values mediates and harmonizes both kinds of value, and causes them to interact in multiplicative ways.

Hence it is suggested that it is a method which has a field of application wherever we find it desirable or necessary to combine technical and culture studies. In this age when new arts and sciences are raining down from the sky so fast that one is kept busy dodging them to save himself from destruction, if we can discover in each department a certain idealistic toolage, master this in one department and use it in others, it will be the source of much needed economy and uplifting power. It may be, therefore, that the method here suggested is a step toward meeting some pressing educational needs in their larger aspect.

II. TIME OF INTRODUCTION AND LIMITATIONS

J. MELVILLE MCPHERRON, HEAD OF DEPARTMENT OF MATHEMATICS,
HIGH SCHOOL, LOS ANGELES, CAL.

I am not expected to make any argument in favor of the value of original work in geometry. While some may dispute that the power of reasoning developed in the study of mathematics is available in other subjects, perhaps all will agree that the reasoningpower developed in the proper study of geometry is available in other branches of mathema

tics.

Frank A. Hill, of Boston, in the Educational Review, some years ago said:

One peculiar advantage of right mathematical work lies in the completeness and accuracy of the results attainable. I am not underrating the value of study in English, in history, or in any of the vast, indefinite and never-to-be-compassed fields. I am simply saying that the demonstration of a theorem in geometry, for instance, may be brought to a kind of finish and completeness impossible in the study of a paragraph about the character of Henry VIII. or the causes of the Civil War, and that the student enjoys a unique consciousness of power in mathematics when he brings a piece of work to a triumphant end. I have noticed that when boys and girls in geometry, for instance, become once imbued with a thinking, investigating, inventive spirit, and with that conception of a proof which gives the child who has it confidence to stand against the world, the subject has a peculiar fascination for them. They work with enthusiasm; the real student glow is there; the inspiration continues operative away from the special influence of the classroom; and the emotional excitement of the "eureka" when the way has been discovered is hardly equaled in any other student experience.

My own experience and observation compel me to agree heartily with the author just quoted.

When should this original work begin? After many years of experience and, I confess, some experimenting, I am fully convinced that the time for this work to begin is when the study of demonstrative geometry is entered upon. Surely the student is entitled to all the help and inspiration he can get from the beginning.

I talked with one noted teacher and book-maker who advocated the plan of going thru the book, omitting the exercises entirely and then returning to the beginning and making a specialty of the originals. I am satisfied he was wrong on this point. I find it very hard to interest pupils in original work, who have been over the propositions of which the proofs are given in the book. It seems hard for them to appreciate the necessity, or the advantage at that stage of the work of their doing anything themselves.

At first only easy theorems and problems should be offered, and some of these of a practical nature so as to enlist the interest of those pupils who do not take naturally to pure geometry or to reasoning at all—and their name is legion. Put a little romance into some of the problems. Great patience and sympathy are required in the beginning with the average pupil. The teacher should put himself in the attitude of the investigator along with the pupil. Let it be understood that we are seeking the solution.

If you have never tried it I think you will find it much easier to stimulate interest in this way than to assume the attitude of being perfectly familiar with these things.

At a very early stage in the work pupils should be asked to give complete proofs without the use of paper or blackboard. It is interesting as well as surprising to notice how some will draw the figure, letter it, and give the demonstration as a purely mental exercise. I have found this exercise helpful in securing the attention of the class. Frequently the one reciting will be corrected for misplacing a letter, or something of that kind.

I think it is a great mistake to suppose that all valuable work done in geometry must be original with the pupil.

A glance at the textbooks in use 50 or 60 years ago will show that original work had no place in the course of study and apparently it had none in the mind of the teacher.