definitions and principles are expressed in the same language which is used in arithmetic and algebra, as nearly as the nature of the case would admit. After the truth of each proposition is demonstrated in relation to magnitudes, it is then verified by numbers. In the notes and illustrations at the end of the volume, and occasionally interspersed through the work, it has not been our design to indulge in abstract discussions or speculations, but to elucidate the subject in question ; not to puzzle the student with difficult demonstrations, but to suggest to him such principles and facts, as will assist him in understanding more clearly the nature and application of the science. We would simply add, in conclusion, that in making these changes and additions, it has been our constant aim to preserve the spirit and character of the original; and while it has been necessary to compress it somewhat by omitting some of the more difficult and least useful propositions and simplifying others in order to bring the work within the comprehension and means of every individual, we have retained all the principles which are likely to be of any practical utility to the learner. We hope these objects have been accomplished in a manner that will be acceptable and satisfactory both to teachers and the public. New Haven, June, 1844. SUGGESTIONS ON THE MODE OF STUDYING AND TEACHING GEOMETRY. I. The first object of the learner should be, to make himself perfectly familiar with the definitions. When he comes to the propositions, the inquiry will arise, what is the import or object of the given theorem, or problem? What is given or admitted as true ; what is asserted merely, and proposed to be demonstrated ? To ascertain these points, it is necessary to read the enunciation with care, at the same time tracing in the diagram the lines and angles to which it refers. His next inquiry is how the truth of this assertion is to be demonstrated; what are the reasons or proofs, which establish its correctness ? Here much time and energy are often wasted, and a lasting disgust for the science imbibed, by attempting “ to commit to memory" the language of the demonstration given in the book. This would be an endless task, and, when accomplished, would be worse than useless. How then is he to demonstrate the proposition, if he is not to learn the language of the author ? It is proper for the pupil to read the demonstration attentively, and, tracing out the several steps in the diagram, fix the leading points in his mind. He should then draw the figure on his slate, or a piece of paper, and go over again with the demonstration, referring to his book only when necessary to learn the order of the steps. This process should be repeated, till he becomes familiar with the train of reasoning, and can go through with it, in his own language, with facility. Finally, laying aside his book, diagram and all, it is recommended to him to form a conception of the figure in his mind, and thus go through with the demonstration independently of extraneous helps, assigning in full the reason for each successive step. He is now prepared to deduce from the particular demonstration the general principle, which of course he will fix indelibly in his mind. At first, this mode of proceeding may be somewhat slow and tedious, but a little practice will give the student surprising facility, and render it a delightful exercise ; while the power of concentrating his attention, the command of language and the accelerated progress thus acquired, will reward him a thousand fold for his first outlay of time and toil. II. With regard to the mode of teaching geometry, it should be constantly kept in mind, that it is not the object to teach words, but principles ; and the amount and kind of instruction, while skillfully adapted to the particular wants and capacities of learners, should be made to bear on this point. The most thorough and successful way of conducting a recitation, is to let the scholar draw the diagram upon the blackboard, leaving his book at his desk; and, if there is any danger that he has prepared his lesson mechanically or by wrote, let him vary the lettering from that of the book. It is not the design of recitations, in this or any other science, simply to hear the pupil “say” his lesson ; but to ascertain whether he understands it; and, if not, to apply the proper remedy. A few such interrogations as these will settle this point at once. Is the proposition a theorem or a problem? What are the points given or assumed to be true? What is the precise point to be proved ? Is the demonstration direct or indirect ? What is the difference between a direct and indirect demonstration ? Is the proposition the converse of any previous one ? What is the meaning of converse, &c. The judgment of the pupil, with the assistance of the notes, will enable him, if he has done justice to his lesson, to answer these and similar questions with correctness and promptitude. Finally, it is recommended to encourage the class to study out different modes of demonstrating the propositions. This is a most excellent exercise to develope their powers of invention and to establish habits of independent thinking and reasoning. Those who have been accustomed to teach Euclid's Geometry, or any other treatise, will find not the least difficulty in using this work. It explains itself ; no principle is employed, till it has been defined, or proved; and when quoted, reference is made to the place where it may be found. Hence, one that is entirely unacquainted with the subject, may ta it up, and easily keep in advance of his class. Circles and the Measurement of Angles. The Circle ; its Diameter, Chords, Areas, &c., The Measurement of Angles, &c., 16 Prop. 13Cor, of 26 Sch. of 13 20 18 Book I. Book I. Prop. 5 & 6B. VI. 6 Sup. I. 3 8 16 9 Sup. I. 4 A. 6 15 11 5 11 3 4 Book VI. Sup. II. 2 1 16) 18 2 17 3 3 E. 4 4 F. Cor. 2. of 4 11 7 7 Cor. of 7 6 9 12 10 14 2 11 Cor. of 13 13 9 15 16 16 17 18 2 19 20 20 A. 15 & 16|B. VI. 2 15 Sup. III. 15 1 18 4 2 10 6 7 8 24 9 12 Cor. 1. of 12 20 Cor. 3 of 12 12 31 13 14 22 14 15 9 Cor. 1, 2. 15 Cor. 1, 2. 15 Book V. Sup. B. I. 1 Prop. 1 Sup. III. 18 21 13 5 16 21 66 30 |