iv This method seems to have been adopted to avoid the difficulty which beginners experience in comprehending abstract propositions. But in avoiding this difficulty, and thus lessening, at first, the intellectual labour, the faculty of abstraction, which it is one of the peculiar objects of the study of Geometry to strengthen, remains, to a certain extent, unimproved. Geometry is a train of connected principles. Its axioms are abstract truths, to which the mind by the very law of its nature readily assents. The existence of these truths is independent of lines, or figures'; and to illustrate them by diagrams, would rather limit than extend our ideas. The propositions of Geometry are also general truths, and co-existent with extension.' In enunciating them, therefore, there seems to be no good reason for limiting their application to the particular diagrams presented to the eye. Geometry is not studied merely for the facts which it teaches--merely because it shows certain relations existing between bodies, and certain properties belonging to them-but, because it disciplines the untrained intellect, and conducts the untaught mind to the temple of truth. The study of Geometry ought, therefore, to be so pursued, as to improve that faculty of the mind which enables it to comprehend general propositions, and to pursue trains of thought disconnected with sensible objects. These considerations have induced the Editor to venture the alterations he has made, notwithstanding that the other method has been followed by the eminent author and his distinguished translators. In the Trigonometries, the Editor has taken the liberty to omit several of the articles a few also have been added. The Author will perhaps not feel himself responsible for that part of the volume, in its present form. MILITARY ACADEMY, West Point, August, 1828. CONTENTS. Book II. The circle, and the measurement of angles, 26 Problems relating to the two first Books, Book III. The proportions of figures, Book IV. Regular polygons, and the measurement of the Book V. Planes and solid angles, Book VIII. The three round bodies, Appendix to Books VI. and VII. Notes on the elements of Geometry, Division of the circumference, General ideas relating to sines, cosines, tangents, &c. - 259 Theorems and formulas relating to sines, cosines, tan. On the construction of Tables, Principles of the solution of rectilineal triangles, Solution of rectilineal triangles in general, Examples of the solution of rectilineal triangles in general, 286 Principles for the solution of right-angled spherical triangles, 291 Solution of right-angled spherical triangles, Principles for the solution of spherical triangles in general, 298 |