same old way. The methods have not changed sufficiently to meet the new conditions. It is right that the logical aspect of the subject should not be neglected. We teach geometry largely as a training for the mind, but in this we must remember a psychological question is involved. Young boys and girls should not study geometry as was done by mature minds centuries ago. But that is essentially what we are asking them to do when we try to adapt students' minds to the logical demands of geometry, instead of adapting the subject-matter of geometry to the minds of the students, which is the demand of psychology. The proper pedagogical procedure is to secure the right balance. between the psychological and the logical. Develop the mind logically but consistently with its needs and demands. Some minds take to the logical naturally, and hence a strict demonstrative system of teaching affords the proper training. But abstract reasoning is generally enjoyed only by the few. The majority of pupils are more interested in practice, particularly if this has a motor phase, and this interest should be appealed to. Pure logical work is heartily disliked by many students, especially girls. As Professor Dewey suggests, there may be more "ultimate logical worth" in merely doing things than in thinking along lines that are repugnant.1 The efforts being made by a few teachers to make the teaching of geometry experimental' is a move in the right direction. Such a system should be able to preserve the logical value of the subject and, at the same time, recognize the psychological principles involved. We have teachers who have employed the "heuristic" method, in which the pupil works out everything for himself. It has an inductive character and in that sense is experimental. Such a method has value, but particularly so when the proper problems arise which act as incentives to stimulate logical inquiry. This is now practiced somewhat in the upper grades of our elementary schools, but it is almost unknown in our high schools. Even without the aid of physics and astronomy, the principle can be carried out. The beginning work in the high school should be independent of any text-book. The 1 Dewey, The Psychological and the Logical in Teaching Geometry, Educational Review, 1903, pp. 389-399. 2 Used here synonymously with inductive work, including the use of some forms of instruments. teacher and pupils should develop the work independently of any such restriction. The use of the scale, protractor, and triangle is as necessary as the Euclidean rule and compasses. The work can be made experimental in two ways, either in an indirect way by testing the truth of suggested theorems and seeking to discover further relations, or better, by associating the theory with some field problems. Every school which can afford it should have a transit for surveying purposes; or better still, a plane-table made by the students will give admirable service. The field problem can be easily chosen to include some proposition in the class work. The habit of performing these experiments after the theory has been learned is not the best plan. The experiments should precede the theory, so as to give an incentive for logical study. Even after the logical work is well advanced, this experimental work should not be abandoned. In this field work proper notes should be taken and careful drawings made to scale. We recall that Archimedes proved propositions in two ways, by pure geometry and by experimental methods. He compared, for example, the areas of two plane figures under the supposition that they had equal weights. Such a method is entirely in accordance with good pedagogy to-day. The use of squared paper is to be recommended in checking up the theorems in mensuration. All work that has thus far been called experimental becomes real laboratory1 work, as the term is used in science, where individual and not class work predominates. The class hour in the United States is generally used for the "hearing" of lessons. In Germany and France, on the contrary, the work is developed in the class-room under the guidance of the teacher. Logical training is the aim and it is rigidly adhered to. In Germany especially, the text-book is consulted after the lessons have been developed. With us, lessons are assigned in advance. In Germany, the class as a unit takes part in demonstrating a theorem. With us, the individual receives the attention. Good teachers have used this principle of individualism to advantage by allowing certain interested students to carry on special investigations. A proper balance between 1 Professor Safford in 1888 recommended the teaching of mathematics by the laboratory method. See Safford, Mathematical Teaching and its Modern Methods. the extreme individual and class methods should produce the best results. With respect to method we may conclude that: a. We need the various methods of attack given us by the Greeks. The use of intersection of loci and of analysis is of especial value. b. The method of question and answer is of value when not used dogmatically. c. The inductive method should precede and accompany the demonstrative. d. Practical experimental work gives rise to proper incentives for logical investigation. e. The first work in geometry should be independent of any text. f. The class hour should be a time for investigation rather than for the "hearing" of lessons. g. The best features of the individual and class methods should be maintained. CONCLUSION The development of the geometric consciousness of the race is duplicated to a marked degree in the mental growth of the child. Recognizing that the child should study practical and intuitive geometry before the logical, we should distribute the work in geometry throughout the school years to conform to this development. Hence the work in the first school years should be a study of geometric forms. This should be comparative in its nature and out of it should be developed the idea of measurement. The purposefulness of measurement is brought out in mensuration when the pupil has had sufficient work in arithmetic. Experimental geometry finds a fitting place in the mensuration of geometric figures, at which time the pupil becomes familiar with the various instruments of construction. The way is thus paved for an inductive geometry, which is the fitting propedeutic for the deductive study in the secondary school. In a general way, the race has passed through similar stages to reach the plane of geometric logic. We may say that the student has reached the Grecian stage when he begins the study of deductive geometry. Notwithstanding that Euclid was not generally adhered to in succeeding centuries, and notwithstanding that geometry was to a certain extent taught with reference to its applications, much of our secondary teaching to-day makes the pupil live entirely in the Grecian age. There is need of a wider recognition of the unity of the mathematical subjects, in particular with reference to their common applications. The teacher who desires to reform the teaching of geometry should remember: 1. That the mathematical subjects should not be taught in isolation from one another. 2. That modern synthetic geometry should have some claim for recognition. 3. That the race preceded the study of logical geometry by the practical. 4. That the history of the teaching of geometry shows that in succeeding centuries logical geometry has been taught gradually to younger students and that modern practice does not sufficiently recognize this fact. 5. That the psychological as well as the logical aspect of the teaching of geometry should be considered. 6. That inductive geometry should precede and to a certain extent accompany the deductive study. 7. That the teaching of geometry should have an experimental character. BIBLIOGRAPHY ORIGINAL SOURCES Anglicus, Robertus. Der Tractatus Quadrantis, in deutscher Übersetzung aus dem Jahre 1477, herausgegeben von Maximilian Curtze. Abhandlungen zur Geschichte der Mathematik, 1899, pp. 43–63. Anonymous. Geometria deutsch, 15th century, Germany. In Arnauld, Antoine. Nouveaux élémens de géométrie, Paris, 1683 (2d ed.) Bartoli, Cosimo. Del modo di misurare, Venice, 1589. Belli, Silvio. Libro del misurar con la vista, Venice, 1569 (1st ed. 1564) Bézout, Etienne. 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