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PLANE AND SOLID

GEOMETRY

INDUCTIVE METHOD

BY

1857-

A. A. DODD, M.S.D., S.B., AND B. T. CHACE,
Teachers of Mathematics in the Manual Training High School,

Kansas City, Mo.

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PREFACE.

The purpose of this elementary treatment of Plane Geomelry is to give to the pupils of the Kansas City Manual Training High School a course in geometric reasoning based principally on the Inductive method of presentation.

In the beginning, by many questions and suggestions carefully arranged, the pupil is led to grasp some of the fundamental ideas of Geometry, and in the same manner the first propositions are established.

Instead of giving the formal proposition at the beginning of a demonstration and then the proof, the pupil reaches the general truth as a result. This result or proposition is to be neatly written and numbered on the blank pages at the back of the book. With the lessons prepared, partly by answering the questions, and under the direction of skillful teachers, it is hoped that the majority of the pupils will be able to make logical deductions from data given and to become selfreliant. It is not the amount of knowledge possessed, but rather the method by which the knowledge is gained, which is the important thing. What power results from the investigation of the truths in Geometry is the all-important question.

No boy or girl will learn to ride a bicycle by memorizing the most carefully prepared directions. Actual struggle and persevering efforts with thoughtful direction bring skill to the learner.

No amount of memorizing of reasoning processes will make a pupil proficient in reasoning. It will tend to keep him from using his originative and reasoning powers.

It is supposed that the pupils who take this course have had, at least, one term in Inventional or Constructional Geometry; if the class has not had this preliminary work, the teacher should spend some time in introducing the subject

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concretely, familiarizing the pupils with dividers and rule; to pupils thus prepared the construction of most plane figures and the grasping of simple geometric ideas should present no serious difficulties. The energies of the pupil can be directed to the processes of reasoning. In this school especially do we try to teach by doing, guided by skilled directors. What the actual experimental work in the laboratory, shop, or cooking-room is to the theory of the science taught, so is the original solution or demonstration to the principles and theorems in Geometry.

For this reason, numerous graded exercises are given along with the propositions. Many of these exercises are intended to make the pupil feel that Geometry is a science which lias to do with common affairs. An early introduction of the properties of the triangle is easily made to the pupil by the method of superposition; and the utility of this method is of great value in acquiring other geometric truths.

Accuracy, neatness in demonstration, and the giving of authority are insisted on in all the work to be done. Independent solutions are encouraged.

Nearly every English Elementary Geometry has been made use of in securing suggestions and hints on demonstrations, but the method of reaching the general truths of Elementary Geometry is unique and is believed to be on the laboratory method of teaching. It must not be forgotten that those who take this course should have finished a book like Spencer's Inventional Geometry, or Nichols' Constructional Geometry, or Hornbrook's Concrete Geometry, or to have had a good preliminary introduction to Demonstrative Geometry. In most instances the question whose answer is the proposition required is printed in pica, thus helping the pupil to keep the main question in mind.

A. A. DODD.

B. T. CHACE. Manual Training High School, Kansas City, Mo.

SUGGESTIONS TO TEACHERS.

It is of the greatest importance that the pupil clearly understand the status of his work in Constructional Geometry. Of the three books mentioned in the Preface, Nichols' is the only one which attempts pure demonstrations. But even these cannot be accepted (see pages 32, 41, 48, 54, etc.), since the problems, (1) To construct an angle equal to a given angle; (3) To bisect a given angle, etc., have not yet been proved to be geometrically true. In Proposition I. he does not know that he has two triangles in which two respective sides and included angles are equal because he constructed them equal, but because they are so given in the hypothesis.

In beginning the formal demonstration of Proposition I. the teacher should clearly outline the method of proof and by numerous exercises and illustrations see that this is fully understood by the pupils. Let the pupils understand that in Proposition I. and others the questions asked about the figures are no part of the demonstrationthey are asked simply to lead the pupil to discover that proof and the steps in it which he is required to give in logical order.

ORDER OF PROOF.

I. General Enunciation, which should be clearly shown to consist of two parts, the hypothesis (or supposition), and the conclusion. Thus in Proposition I. we have: Hypothesis.-If two sides and the included angle of any triangle are equal, respectively, to two sides and the included angle of another triangle, conclusion, the triangles are equal.

II. The Particular Enunciation, which refers to the particular figure or figures which fulfill all the given conditions of the General Enunciation. Thus in Proposition I.:

Given: The triangles A B Cand DEF, in which AB, AC,

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