scholarships and other prizes, each student strives to outdo his fellow. In our own country we are between these two extremes, but lean toward the development of the individual.

In answer to the question why geometry has been taught, two reasons may be mentioned. It has been taught for its practical and logical values, and of these the logical has stood out the most prominently. Before the sciences of the present day entered the schools, the classical and mathematical studies afforded the supposed requisite mental discipline. And it was and still is to a great extent believed that this training gives to the student a certain formal power which he can apply in other fields, and our methods have been shaped accordingly.

THE EDUCATIONAL SITUATION TO-DAY

We are living in an age which demands action. It is no longer sufficient that one be able to think logically, but one must be able to do things as well. The demands of society are so complex that the finishing schools must train along many lines and not, as in the days past, train simply for the "higher callings." So we have many special schools that train for various vocations. In Germany the public schools seem well adapted to meet the present conditions. There, the boy, if he must earn a livelihood as soon as possible, attends the elementary school (Volkschule), or at most the high grade elementary school, or perhaps the Realschule. The Realgymnasium and Oberrealschule prepare for the commercial and industrial callings, and the Gymnasium for the "higher callings." In England, also, there are schools that provide for class distinction, but the secondary schools, for the most part, prepare for the universities and the positions to which gentlemen of quality are called. The rise of technical schools in England is in response to a demand for better training for effective service in industrial and technical lines. In the United States we have a few high grade technical schools, but we are behind England in the organization of secondary technical education. Unlike Prussia, our public school system is not well adapted to meet the various demands that are put upon the schools. We have one set of institutions, through which must pass pupils with different aims in life. When the elementary school is finished, one pupil begins to function in the world, another enters the high school.

In Germany, two such boys would be in different institutions from the beginning. In our high school opportunity is given for a selection of courses. In Germany, the boy of high-school age has a choice of several institutions. While the American high schools are beginning to offer certain courses, for example the commercial, that prepare for immediate life work, the pressure of the university is so great that the high school finds it difficult to perform the double function of serving both as a preparatory and as a finishing school.

The new education is demanding a preparation for effective service, whether this be in the elementary school, the secondary school, or in the university. The school subjects should not be taught merely as a preparation for higher studies in the same or kindred lines. They should have a present value for the student. This requires a closer unity between schoolroom practice and the life the pupil is then living, and the problem as it relates to geometry will now be considered.

PRESENT PROBLEMS IN THE TEACHING OF ELEMENTARY GEOMETRY

The teaching of geometry has been concerned primarily with its subject-matter and its logic. There has been the field of geometry into which the learner enters. The aim has been to acquaint him with this field. In learning its content according to prescribed rules, the mind of the student is trained in logic. The student has been adapted to set standards. One thing that has not been done is to recognize that each student has an individual psychology and that, to meet these conditions, geometry should lose its traditional fixedness and be made adaptive to the needs of individuals. Good teaching will take cognizance of three factors involved. The first consideration is the psychology of the individual; the second, the need of the logical training of the mind; the third, the recognition of the demands of the present age. The history of the teaching of geometry gives us a perspective whereby we can the better give attention to these considerations. It has also a prospective side in helping us to guard against the errors of the past and points out that which has been of value. Certain present problems in the teaching of geometry have a relation to the foregoing historic material. We shall here be concerned with:

1. The balance between the logical and the practical.-Logical geometry is generally associated with the theoretic. The history of the teaching of geometry points this out. Logic did not enter the practical work of the Egyptians, while the Greeks on the other hand did not admit practice in the theory. In later times, we have seen that practice has entered into the school work, but at most it has followed the theory, not served as an introduction to it. Thus logic has been disassociated from practice. When theory, however, is approached inductively, opportunities are offered for practical logical work. We are so accustomed to use in geometry the term logic for deductive logic that we have almost forgotten that inductive logic has a place in teaching. From a series of concrete experiences, the pupil may be led, after the inductive method, to formulate some general principle. And the work can be made practical by employing the scale, the protractor, the compasses, and instruments for field work, not to mention the related numerical exercises. This inductive work forms a fitting propedeutic to the ordinary high school course in geometry, and finds a fitting analogy in the history of geometry where the practical stage preceded the logical. Even after deductive geometry is begun, the inductive treatment could well be continued as a method of discovery.

Practical field problems serve to point out the existence of certain theorems, which should then be logically proved. We thus see the need of a continual co-operation between theory and practice. The latter serves to verify the former and many times to check erroneous conclusions. This is seen in the practice of drawing accurate figures. Such a plan serves not only to check illogical conclusions, but frequently to suggest a line of proof by mere observation of the relations between the lines or angles drawn.

The history of the teaching of geometry tells us that the mind tends to hold to the special and approaches the general with effort. We have seen this in the Greek development and especially in the practical geometries of later times. We should give heed to this tendency in our teaching and not try to force too early the idea of the general. This argues here again for the inductive method in the first stages of the work. As a result of following to the extreme the deductive method, we find our geometries stating the general theorem, followed by the proof

and a list of corollaries. At least some of these could well be used as special cases which lead up to the general theorem. This of course does not apply where the general theorem is of a very simple nature.

Our history also tells us that many of the geometries in use in the past have employed fallacious proofs. This has already been pointed out in certain French and German texts. Perhaps the most common example of this was in connection with the theory of parallels. Some of the simpler theorems were sometimes stated without proof. Has this any significance for us in our schoolroom practice to-day? Are we to require all students to master the most difficult propositions? Are we to assume any theorems without proving them, as Professor Perry suggests? It would be impossible to answer these questions in the light of history. That such practices were common is known. France and Germany have produced more famous mathematicians than has England, which has closely adhered to the logic of Euclid. But if France and Germany had adhered closely to Euclid, would they have produced still a greater number of famous mathematicians? The question must remain unanswered. If the teacher desires to defer a certain proof on account of its difficulty, or to take for granted some theorems seemingly selfevident, he certainly is following historic precedent. It seems wise to assume, at least tentatively, some of the most simple propositions. For the beginner, these are the hardest to understand. The theorem that all right angles are equal was taken as axiomatic by Euclid, but modern books give it as a theorem. It is capable of proof, but when proved with all the "machinery" that accompanies a proof, it becomes difficult for the beginners. It is better to test the truth of the statement by paper-folding, and then set up the theorem as one to be quoted. As an example of theorems that could well be deferred, all teachers recognize the difficulties in presenting the incommensurable case in the applications of the theory of proportion to various magnitudes. The texts used in France and Germany up to the nineteenth century, and perhaps later, usually neglected this case entirely. There may be a question as to what extent texts can be arranged to accommodate these breaks in logic, but the teacher does not depart from precedent if he defers any theorem or group of theorems for later treatment.

Practice left unguided by theory is apt to lead to erroneous results. The Egyptians used incorrect formulas for certain rectilinear areas. Students of geometry to-day often adopt these practices. It is the province of theory to show wherein the error lies and point to the correct solution. But theory often takes the concreteness out of things. The fact that a boy uses an erroneous formula of his own construction does not by any means indicate a low order of thinking. The great Gerbert, who knew little of Euclid, found incorrectly the area of an isosceles triangle. The boy who has erred as described has used his judgment, which is not always the result of a training in logic. In rectifying such errors the pedagogic teacher will seek to stimulate the right growth of such judgment and not replace it by a blind adherence to the results obtained by logical reasoning. In seeking a balance between the logical and the practical, we may observe that:

a. The general should be approached through the special in early work. Practical tests have a place here in serving to check theory. Such work may lead to discoveries.

b. Theory is a guide to practice.

c. Logic need not be divorced from practice. Inductive geometry has a practical aspect.

d. Induction should be employed even after the deductive study is begun.

e. There is historic precedent in assuming self-evident theorems and such as require a very high grade of reasoning.

2. The sequence in the subject-matter of geometry. A sequence that is logically best is not necessarily pedagogically the best. The logical sequence is a thing that concerns the science of geometry. It is a passive, immovable thing. The pedagogic sequence-if we could have one-concerns the minds and aptitudes of the pupils and the demands of present living. It must be alive and subject to change. So a geometry constructed to meet the needs and demands of the student must necessarily be different from one that seeks alone to perfect a system of logic. Teachers of mathematics above all others have been slow in recognizing these distinctions. Text-books, beginning with Euclid, have, to a large extent, been written from the standpoint of the development of the science rather than from that of the needs of the pupil.