Many of the exercises at the foot of the page require the student to infer the truth involved in the relations given. The interrogative form is employed for the purpose of compelling the student to obtain the ideas for himself, and the answers he must give to the questions furnish an admirable training in accuracy of expression. A great abundance of undemonstrated theorems and of unsolved problems is supplied, and teachers will find them quite numerous enough for the needs of any class. The demonstration of original theorems and the solution of original problems are of so great consequence in developing the power to reason that every teacher should insist upon such work. Much aid in originating demonstrations may be obtained from the Summaries which follow each of the first six books. These summaries are not collections of propositions that have been demonstrated, but are rather groups of the truths established in the book to which they are appended. If the student makes himself thoroughly acquainted with them, much of the difficulty experienced in demonstrating original theorems, in solving problems, and in determining loci will be removed. A very small proportion of those who study elementary geometry expect to become mathematicians in any broad sense of the term, and so geometry must serve to give them almost the only training they will get in formal and logical argument in secondary schools and in colleges. For this reason mathematical elegance in demonstrations and in solutions has often been sacrificed in the interest of clear and simple steps, even though such a plan has required some expansion of the text. Elegant demonstrations are appreciated by mathematicians, but training in formal deductive reasoning is of more consequence to most students. The author is indebted to many authors, both American and foreign, who have preceded him. Their efforts to present the subject in the best way have aided him very greatly in preparing this work. He has selected large numbers of supplementary theorems and problems from several European authors of renown, and yet he is unable to give credit to any author in particular, because they all seem to have selected their exercises from some common source of supply. ALBANY, N.Y. WILLIAM J. MILNE. SUGGESTIONS TO TEACHERS 1. Thorough teaching and frequent reviews, especially at the beginning of plane and of solid geometry, will be rewarded by intelligent progress and deep interest on the part of the students. 2. Before the assignment of any lesson, the teacher should require the students to draw the figures and answer the questions which are introductory to the propositions that are to be proved at the next lesson. After the questions have been answered, require the students to express their inferences in the form of a theorem. 3. While the students are answering the introductory questions or stating the inferences suggested by the exercises at the bottom of the page, the inquiry, "How do you know that this is true?" will often lead to a demonstration. 4. The section numbers are convenient in written demonstrations, but in oral proofs the reason for each step should be given fully and accurately and all why's should be answered. 5. Students may sometimes be allowed to express definitions, axioms, theorems, etc., in their own language, but as a general rule their expressions are inaccurate and faulty. The teacher should in such instances call attention to the errors and require concise and accurate statements. It will then be discovered that they approximate very closely those given in the book. 6. The practice of requiring the students to outline, in a general way, the steps they are to take in establishing the truth of a proposition will develop much logical power and cause them to look at the argument rather than at its details. The following are suggestive outlines of steps: Prop. XXXVI., page 61. 1. Draw the diagonal AC. 2. Prove & ABC and ADC equal. Prop. XII., page 185. 1. Make the required construction, drawing CE, BJ, and CK. 2. Prove A AEC and AJB equal. 3. Prove AEKL≈2 ▲ AEC, and ACHJ≈2 ▲ AJB. 4. Prove AEKL≈ ACHJ. 5. Similarly BDKL≈ BCGF. Then, ABDE≈ BCGF+ ACHJ. If suggestion 6 is 7. Demonstrations should never be memorized. observed carefully, students will not be likely to commit to memory the words of the book. 8. Encourage students to prove propositions in their own way, even though the proofs be less elegant than those which are given. Elegant methods will be acquired by practice. 9. Written demonstrations should be required frequently. They serve a double purpose, viz.: they train the eye and develop accuracy in reasoning. All written work should be done neatly, and all figures should be drawn as accurately as possible. 10. The undemonstrated theorems and unsolved problems are probably more numerous than most classes can prove or solve in the time allotted to the subject, consequently teachers are expected to make selections from the lists given. The exercises are carefully graded so that the more difficult ones come at the end of each list. These may be omitted at the first reading and reserved for a final review. It is suggested that the exercises in the interrogative form at the foot of the page in Books I and II, except the numerical ones, be employed at first only for the purpose of developing correct geometrical concepts and accuracy in expressing the truth inferred. In review the proofs of the inferences may be required. 11. Particular attention should be given to the Summary at the end of each book. The students should be required to state all the conditions under which the facts given in black-faced type have been shown to be true. They will thus have at immediate command all the facts which can be employed in the demonstration they are attempting. If the demonstration of the inferences and theorems found at the bottom of the page is required, the students should be referred to the summary. They should understand, however, that they can use no truth given in the summary whose section number indicates that it was established subsequently to the point in the text where the proposition or exercise is found. The method of using the summaries is illustrated upon page 78. |