Solidity of the Frustum of a Pyramid, - Solidity of a Rectangular Prismoid,.. Surface of a Frustum of a Cone,. ELEMENTS OF GEOMETRY. INTRODUCTION. 1. SPACE extends indefinitely in every direction and contains all bodies. 2. EXTENSION is a limited portion of space, and has three dimensions, length, breadth, and thickness. 3. A SOLID, or BODY, is a limited portion of space supposed to be occupied by matter. The difference between the terms, extension and solid, is simply this: the former denotes a limited portion of space, viewed in the abstract, while the latter denotes such a portion occupied by matter. The term, Solid, is generally used in Geometry, in preference to Extension, because the mind apprehends readily the forms and relations of tangible objects, while it often experiences much difficulty in dealing with the abstract notions derived from them. It is, however, important to observe, that the geometrical properties of solids have no connection whatever with matter, and that the demonstrations which establish and make known those properties, are based on the attributes of extension only. 4. A Solid being a limited portion of space, is necessarily divided from the indefinite space which surrounds it that which so divides it, is called a Surface. Now, since that which bounds a solid is no part of the solid itself, it follows, that a surface has but two dimensions, ength and breadth. 5. If we consider a limited portion of a surface, that which separates such portion from the other parts of the surface, is called a Line. This mark of division forms no part of the surfaces which it separates: hence, a line has length only, without breadth or thickness. 6. If we regard a limited portion of a line, that which separates such portion from the part, at either extremity, is called a Point. But this mark of division forms no part of the line itself: hence, a point has neither length, breadth, nor thickness, but place or position only. 7. Although we use the term solid to denote a given portion of space, the term surface to denote the boundary of a solid, the term line to denote the boundary of a surface, and the term point to designate the limit of a line, still, we may employ either of these terms, in an abstract sense, without any reference to the others. Thus, we may contemplate a river, as a solid, without considering its boundaries; may look upon the surface and perceive that it has length and breadth without refering to its depth; or, we may regard the distance across without taking into account either its depth or length. So likewise, we may consider a point without any reference to the line which it limits. In the definitions and reasonings of Geometry these terms are always used in an abstract sense; they are mere signs to the mind of the conceptions for which they stand. 8. ANGLE is a term which designates the portion of a surface included by two lines meeting at a common point; and it also denotes a portion of space included by two or more planes. 9. MAGNITUDE is a general term employed to denote those quantities which arise from considering the dimen sions of extension, and is equally applicable to lines, angles, surfaces, and solids. Geometry is conversant with four kinds of magnitude. 1. Lines; which have length without breadth or thickness. 2. Angles; bounded by straight lines, by curves, and by planes. 3. Surfaces; which have length and breadth without thickness: and 4. Solids; which have length, breadth, and thickness. 10. FIGURE is a term applied to a geometrical magnitude and expresses the idea of shape or form. It is that which is enclosed by one or more boundaries. Thus, "A triangle is a plane figure bounded by three straight lines." 11. A PROPERTY of a figure is a mark or attribute common to all figures of the same class. 12. The portions of extension which constitute the geometrical magnitudes, are indicated to the mind by certain marks called lines. Thus, we say, the straight line AB, is the shortest distance between the two points A and B. The mark AB, on the paper, is not the geometrical line AB, but only the sign or representative of it-the geometrical line itself, having merely a mental existence. We also say, that the triangle ACB is bounded by the three straight lines AB, AC, CB. Now, the triangle ACB, is but the sign, to the mind, of a portion of a plane. That which the eye sees is not the geometrical conception on which the mind acts and reasons but is, as it were, the word or sign which. stands for and expresses the abstract idea. A A -B These considerations have induced me to represent the geometrical magnitudes by the fewest possible lines, and to reject altogether the method of shading the figures. It is the conception of extension, in the abstract, with which the mind should be made conversant, and too much pains cannot be taken to exclude the idea that we are dealing with material things. |