4-12-23 PREFACE. IN monly known as the Euclidian Geometry, it is aimed to combine the excellencies of Euclid with those of the best modern writers, especially of Legendre, and Rouché and Comberousse. Many of the demonstrations are those of Euclid, with minor changes frequently introduced, and the syllogistic form is retained throughout; but the arrangement is quite different. Many objections have been made against Euclid. His definitions are not all of them the best, nor are they in their proper places. His treatment of angles is deficient. His arrangement of the propositions is often poor, mixing straight lines, angles, triangles, etc., without any regular classification; and his demonstrations are sometimes cumbersome and prolix. Nevertheless, although numerous attempts have been made to improve upon Euclid, it still remains the great model, the unrivalled original, on which is founded the whole system of elementary Geometry. Perhaps a more finished specimen of exact logic has never been produced than that of the old Greek Geometer. In the present treatise it is desired to effect two objects : (1) to teach geometric truths ; (2) to discipline and invigorate the mind, to train it to habits of clear and consecutive reasoning. Accordingly, more numerous propositions have been given, and the demonstrations made more complete, than either object alone would seem to demand. In each proposition is a distinct statement, of what is given, of what is required, and of the proof. Each assertion in the proof begins a new line, and is accompanied by a reference to the preceding principle on which the assertion depends. These references are quoted two or three times in small type, and afterwards referred to only by number. The student should always be ready, if required, to quote the proper reference, and to show its application. The text is so arranged that the enunciation, figure, and proof of each proposition iii are in view together; and notes are directly appended to the propositions to which they refer. A few symbols and abbreviations of words have been freely used, but only such as bave long been employed by mathematicians, and are recognized by the majority of teachers. In the figures the given lines are represented by full lines, those which are added to aid in the demonstration, by short-dotted lines, and the resulting lines by dashed, or long-dotted lines. In solid geometry the dotted lines represent those which a solid body would conceal. The propositions marked with a star may be omitted without interfering with the continuity of the work, but the omission is not recommended. The exercises distributed through the text are quite easy, and may all be worked out by the average student; those at the end of each book are more advanced, and have been carefully graded, with hints appended to many of the more difficult ones. It is only in original demonstration that the student can acquire mental power. More discipline is gained in working out one demonstration, without aid, than by learning a number of them that are given by others. A student can never really comprehend a subject if he only tries to understand and remember what the book says. The subject can become known to him only by his thinking To develop the power of independent thought in the student, is the most important part of the teacher's work, and it is the most difficult. In preparing this work I have consulted some of the best American and English books on Euclidian Geometry, and am especially indebted to the text-book recently published by the English Asso. ciation for the Improvement of Geometrical Teaching. I have also derived assistance from a number of French works, especially from those of Catalan, Briot, and Legendre, while the Traité de Géométrie by Rouché and De Comberousse has been my constant companion. It remains for me to express my thanks to Prof. R. W. Prentiss, of the Nautical Almanac Office, for reading the MS., and to Mr. I. S. Upson, the College Librarian, for reading the proof sheets. Ε. Α. Β, upon it. New BRUNSWICE, 90., Way, 1890. } RATIO AND PROPORTION : SIMILAR FIGURES. Numerical Relations between the Different Parts of a Triangle 158 BOOK IV. AREAS OF POLYGONS. Measurement of Areas.. Comparison of Areas Problems of Construction.. Applications... Exercises. Theorems.. Problems.. 180 187 191 200 201 206 BOOK V. MAXIMA AND MINIMA. REGULAR POLYGONS. THE CIRCLE. Regular Polygons..... The Measure of the Circle.. Principle of Limits..... Problems of Construction. Problems of Computation.. Value of nl. Method of Perimeters. . Maxima and Minima.... Exercises. Thcorems. Numerical Exercises. Problems... Exercises in Maxima and Minima 208 215 216 224 229 231 234 243 245 247 248 |