SUGGESTIONS TO TEACHERS. Geometry is essentially a disciplinary study, and benefit derived from its study depends upon the independent thought expended by the pupil. A geometry is in the nature of a "key"' to the extent to which the demonstrations are written out for the pupil. That parı of the work which a pupil can do for himself should not be done for him. The teacher and text-book should furnish the data and stimu. late thought rather than give him a set form of words which he may repeat entire with or without the ideas which those words should convey. In the following pages are suggestions arranged in logical order which are intended to direct and stimulate the thought of the pupil so that he may work out his own demonstrations. Model demonstrations are given of a few propositions to show tbo student how to work out his own demonstrations, and in what form they should be given. By comparison it will be seen that the answers to the suggestions logically linked together, constitute the demonstration. The suggestions should be studied in their order, for usually each suggestion depends upon the preceding one. The answer to a suggestion should consist of a statement of the relations asked for, together with the authority in full for such statement. To be indifferent in regard to the authority in any instance is to encourage carelessness, slovenliness and inaccuracy in demonstration. A common error is to apply authority that does not exactly fit the conditions udder consideration. The pupil should be made to clearly understand that the authority should, without exception, be a definition, an axiom, or a previously proved proposition. "'It seems so," or, "it looks reasonable," or any expression of judgment will not do. Tho pupil should be encouraged to search out his own authority, even SUGGESTIONS TO TEACHERS. when the authority is quoted for him in the suggestions, using the reference simply for verification. A pride in independent work is a most important factor in securing satisfactory results. In the preparation of the lesson the pupil should write out his demonstration, noting carefully the form of the "models." This will ensure correct form and avoid haziness of thought. During the first few weeks the teacher should scan carefully this written work as well as tests taken in the recitation. The exercises, or at least a part of them, should be taken along with the propositions as they occur, and not be studied all together at the end of a chapter. Time will be saved in the end by starting slowly but surely, passing over nothing that is not clearly understood. Since each demonstration involves previous propositions and definitions, facility in demonstration can be best secured by committing to memory each theoren and definition; for that authority cannot be readily recognized and applied which is only imperfectly in mind. The subject of proportion will prove to be the most difficult part of geometry and it should be very carefully considered. To teach the theory of proportion by means of numbers, and then apply the principles developed to geometric magnitudes and numbers indiscriminately without consideration of the limitations of the various statements is not scientific. Note 3, page 142, should receive careful attention. In reducing the form = m to A = m B, the tendency is to multiply both members of the equation by B. The thought is correct if B is a number, but the process is unthinkable if B is a geometric magoitude. means that A is measured by the unit B, hence to say that A contains B m times is only another way of saying that A equals m times the unit B, or m B. 12 contains 4 three times (42=3) is another form of expression for 12 equals 3 fours (12=3X4). The ex 1 foot pression = 12, means the same as the expression 1 foot = 12 inches. In this connection see Art. 160. CHAPTER I. RECTILINEAR FIGURES, DEFINITIONS. The space conceived to be occupied Definition. A geometrical solid is a limited portion The term solid will bereafter be used for geometrical solid. 2. The boundary that separates a solid from the space As we can think of the surface of a body without in- Definition. Surface is that which has length and |