ELEMENTS OF GEOMETRY AND MENSURATION. PART II. GEOMETRY AS AN ART. 94. GEOMETRY AS AN ART is the practical application of Geometry as a Science', and is sometimes called 'Practical Geometry', by which is meant Geometry in Practice. This practical application consists in doing those things, which in Part I. it has been shewn may be done, according to strictly defined geometrical notions and principles. For example, in the former Part a square was defined to be "a parallelogram, which has all its sides equal and all its angles right angles"; in this Part such a construction is required to be actually made under certain given circumstances. Also, generally, the Propositions demonstrated in the former Part are in this required to be known, and put to use, for purposes of Construction and Design; and that without any respect to order or precedence, such Proposition being always taken, wherever it may stand in the former Part, as we judge will most readily and efficiently serve our purpose in this. ness. 95. In strict Geometry, be it remembered, a point has no magnitude, neither length, nor breadth, nor thickA line also has length only, and neither breadth, nor thickness. And, in practice, the nearer we can bring our points and lines to these definitions the more strictly correct will be the work depending upon them. For, if that, which should be a fine point, be in fact a circle of considerable size, then in measuring from such a point, or in joining two such points by a line, it is obvious that we should be liable to considerable error. In like man PART II. 1 ner, if we make lines broad and coarse instead of fine, then, in the case of such lines intersecting each other, the points of intersection cannot be accurately marked, and therefore plainly any measurements from such points will be subject to error. And so in other cases, where points and lines require to be actually traced. Hence, although perfect accuracy is really unattainable, it is plain, that, in the application of Geometry to practical art or design, correctness of construction is most nearly attained where the precision of the Geometrical Definitions is most closely regarded. 96. But before Geometry can be put in practice, certain Tools or INSTRUMENTS are required, of which we will here give a short description: (1) The POINTED PENCIL, or PEN, or other marker, is used to trace out lines and to mark points on paper, or board, or other surface. It is only requisite for accurate workmanship that the marking point be kept as fine as possible. (2) The FLAT-RULER, or STRAIGHT-EDGE, is used for drawing straight lines on a given plane surface; and for determining whether lines already drawn be straight; and for some other purposes. It is made of various substances, but generally of wood, the only essential requisite being that it shall have one edge, or boundary, throughout its whole length, perfectly straight. This being the case, it is clear that a straight line may be drawn on any given plane surface by placing the straightedge in contact with the surface, and drawing the pencil or other marker carefully along it. And a given straight line may be tested as to its straightness, by placing the straight-edge close along-side the line, and observing whether the two coincide or not with each other. Of course, if the ruler itself be not perfectly straight, it cannot be used to any good purpose, where accuracy of construction is required. But this fault, if it exist, is easily detected by the following simple method: Place the straight-edge in close contact with any plane surface, as paper or board, and draw a straight line along it in the usual way, to the whole extent of the straight-edge. Then turn the straight-edge round so that its extremities exactly change places, and draw a straight line along it again. If the two lines thus drawn coincide throughout their whole extent the ruler is correct; but otherwise not. (3) The COMPASSES consist of two equal legs connected together by a hinge or joint at one end of each, and having the other ends worked down to fine points, which meet closely when the legs are brought into contact, that is, when the compasses are shut. The hingejoint works rather stiffly, so that the legs, when left to themselves, may remain fixed at any angle by which we may choose to separate them. This instrument is used for measuring off short distances, that is, straight lines; and also, when a portion of one leg is moveable, and replaced by a pen or pencil, for drawing small circles. It is obvious, that with such an instrument a circle of any given radius, within certain limits, may be traced. For, if the legs be separated so that the distance between their extreme points is equal to the given radius, then by fixing one point in the paper or board and causing the compasses to revolve round it, the other point, being kept in contact with the paper or board, will evidently trace out the required circle. (4) The SQUARE consists of two flat-rulers firmly connected together in such a manner that both their inner and outer edges are at right angles to each other. This instrument is used chiefly by masons and carpenters for constructing right angles, and for testing the correctness of angles which ought to be right angles. Whether the square itself be correct or not, may easily be determined by the following method: |