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for example, the method of proving the equality of lines by means of equal triangles; the method of proving the proportionality of lines by means of similar triangles, etc., and finally the analyses of theorems and problems are introduced, but in a more concrete form than usual.

The propositions are arranged with the view of obtaining a perfect logical and pedagogical order. An unusually large number of exercises, selected with care for the purpose of securing increased mental power, is given.

The general plan and preparation of the greater part of the book are the work of Dr. Schultze, while that of Dr. Sevenoak has been chiefly editorial.

SUGGESTIONS TO TEACHERS

1. Students should be made thoroughly familiar with the definitions, especially in the beginning. Many beginners who are otherwise perfectly capable of understanding logical deductions, fail in geometry because they have not acquired this thorough knowledge of the fundamental concepts. Exercises in geometrical drawing (e.g. §§ 113, 114, 115, etc.) and numerical exercises afford an excellent means of familiarizing the student with the definitions.

2. The preliminary propositions (§§ 52-58) on account of their great simplicity are quite often confusing for the be ginner and should not be made the basis of the study of geometrical form. Later on, however, too much emphasis cannot be laid upon this form in which demonstrations are presented.

3. Do not omit the exercises at the first reading, although it is not necessary at this time to study every exercise given

in the text.

4. It is advisable to explain at length to the beginner the meaning of the very first exercises, since many students need some practice before they can understand geometrical language.

5. Many beginners experience considerable difficulty in trying to remember the data of a proposition; they forget the hypothesis. Graphical methods afford an excellent means to overcome this difficulty. Draw equal lines or equal angles in the same color, or if this is impracticable, mark them by equal

viii

SUGGESTIONS TO TEACHERS

crossmarks; denote parallel lines by arrows; draw a greater line in some color, e.g. green, a smaller line in another color, e.g. yellow, etc. The following diagrams illustrate the method. The diagrams on the left may be used if no colored crayons are available, those on the right should be drawn in the colors indicated. Lines and angles whose relative size is unknown are drawn in white.

[blocks in formation]

To prove BD = DC and ADL BC (Prop. XVII, Bk. I).

[blocks in formation]

[The hypothesis indicated by either diagram is: In quad rilateral ABCD, AB = CD, and AB|| CD.]

[blocks in formation]

To prove the opposite sides of the quadrilateral are equal. 6. Discourage mere memorizing. Many students form during the first weeks of study the habit of relying too much upon their memories, a habit which often proves disastrous to their future geometrical work.

7. Figures should be drawn as accurately as possible and as general as possible, i.e. a figure relating to a triangle in general should not be a right or an isosceles triangle, etc.

8. Give frequent written examinations containing some simple original work.

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