whose method on this subject is the most simple and logical. Upon the whole, the demonstrations have been put in that form which was thought best suited to the comprehension of beginners having no previous knowledge of mathematics beyond arithmetic. In the divisions of the work, the writer has aimed to follow the simple and natural classification of the subjects treated. The arrangement adopted has enabled him to set down as simple corollaries many propositions which are usually demonstrated separately. The definitions have been distributed between the various sections, instead of being crowded together at the beginning. After each Section, or Book, there will be found a few practical illustrations and exercises. In a Supplement, some examples have been given of the application of Algebra to Geometry. The amount of Geometry contained in the work is sufficient to prepare the pupil for the study of Plane Trigonometry and Surveying. INTRODUCTION. SECTION 1.-GENERAL DEFINITIONS. 1. GEOMETRY is that branch of mathematics which treats of magnitudes in space. 2. A LINE is that magnitude which has length, without breadth or thickness. 3. A SURFACE is that which has length and breadth, without thickness. 4. A SOLID is that which has length, breadth, and thickness. 5. A POINT has position without extension. 6. A STRAIGHT LINE is one which has the same direction through all its consecutive parts. 7. Of BENT LINES, that which is composed of straight lines is called a broken line; that of which no part is straight is called a curve. 8. A PLANE is a surface in which if any two points whatever be taken, the straight line joining them will lie wholly in that surface. 9. MAGNITUDES of EXTENSION are lines, surfaces, and solids. The relative positions of these give rise to magnitudes of direction, or angles, which will be defined in their proper connections. SEC. II.-OF TERMS AND SIGNS. 1. An AXIOM is a self-evident truth. 2. A THEOREM is a statement of a truth which requires to be proved, or demonstrated. 3. A PROBLEM is a statement of something required to be done, or solved. 4. The term PROPOSITION may be applied either to a theorem or to a problem. 5. A COROLLARY is an obvious consequence from something that precedes. 6. A SCHOLIUM is some repark relating to what precedes. 7. The term HYPOTHESIS denotes the supposition made, or the conditions given, in any proposition. 8. The term INFINITE, as used in geometry, means beyond measure; that is, either absolutely beyond limits, or beyond all appreciable limits. 9. A RATIO is the relation which one quantity bears to another, as equal to it, greater, or less. The value of the ratio is the quotient arising from dividing one of the quantities by the other. If A and B represent any two quantities, the ratio of A to B may be written A either A : B, or B 10. A PROPORTION is an equality of ratios. Thus, if the ratio of A to B be equal to the ratio of C to D, those two ratios will constitute a proportion, which may be written A:B::C:D. 11. The sign = denotes the equality of two quantities between which it is placed. IIence, A C A proportion may also be written B D |