But still a large class of seminaries remained unsup plied with a suitable text-book on Elementary Geometry and Trigonometry: viz., those where the pupils are caried beyond the acquisition of facts and mere practical knowledge, but have not time to go through with a ful course of mathematical studies. It is for sen, that the following work is designed. I has been the aim of the author to present the striking and important truths of Geometry in a form more simple and concise than could be adopted in a complete treatise, and yet to preserve the exactness of rigorous reasoning. In this system of Geometry nothing has been taken for granted, and nothing passed over without being fully de:nonstrated. The Trigonometry, including the applications to the measurements of heights and distances, has been written upon the same plan and for the same objects: it embraces all the important theorems and all the striking examples. In order, however, to render the applications of Geometry to the mensuration of surfaces and solids complete in itself, a few rules have been given which are not demonstrated. This forms an exception to the general plan of the work, but being added in the form of an appendix, it does not materially break its unity. That the work may be useful in advancing the interests of education, is the hope and ardent wish of the authoz. FISHKILL LANDING, May, 1851 GEOMETRY. · BOOK I. DEFINITIONS AND REMARKS. 1. Extension has three dimensions, length, breadth, and thickness. Geometry is the science which has for its object: 1st. The measurement of extension; and 2dly, To discover, by means of such measurement, the properties and relations of geometrical figures. 2. A Point is that which has place, or position, but not magnitude. 3. A Line is length, without breadth or thickness. 4. A Straight Line is one which lies in the same direction between any two of its points. 5. A Curve Line is one which changes is direction at every point. The word line when used alone, will designate a straight line; and the word curve, a curve line. 6. A Surface is that which has length and breadth, without height or thickness. 7. A Plane Surface is that which lies even throughout its whole extent, and with which a straight line, laid in any direction, will exactly coincide in its whole length. 8. A Curved Surface has length and breadth without thick. ness, and like a curve line is constantly changing its direction. 9. A Solid or Body is that which has length, breadth, and thickness. Length, breadth, and thickness are called dimen Definitions. sions. Hence, a solid has three dimensions, a surface two, and a line one. A point has no dimensions, but position only 10. Geometry treats of lines, surfaces, and solids. 11. A Demonstration is a course of reasoning which establishes a truth. 12. An Hypothesis is a supposition on which a demonstration may be founded. 13. A Theorem is something to be proved by demonstration. 14. A Problem is something proposed to be done. 15. A Proposition is something proposed either to be done or demonstrated-and may be either a problem or a theorem. 16. A Corollary is an obvious consequence, deduced from something that has gone before. 17. A Scholium is a remark on one or more preceding propositions. 18. An Axiom is a self evident proposition. OF ANGLES. 19. An Angle is the portion of a plane included between two straight lines which meet at a common point. The two straight lines are called the sides of the angle, and the common point of intersection, the vertex. Thus, the part of the plane included between AB and AC is called an angle: AB and AC are its sides, and A its vertex. B An angle is generally read, by placing the letter at the vertex in the middle. Thus, we say, the angle CAB. We may, however, say simply, the angle A. 20. One line is said to be perpendicular to another when it lines no more to the one side than to the other |