An angle is the opening between two straight lines which meet one another. "When several angles are at E D Fig. 2. A one point B, any one of them is expressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another, is put between the other two letters: Thus the angle which is contained by the straight lines, AB, CB, is named the angle ABC, or CBA; but if there be only one angle at a point, it may be expressed by a letter placed at that point; as the angle at E.' When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight lines are said to be perpendicular to each other (fig. 3). An obtuse angle is that which is greater than a right angle (fig. 4). Fig. 3. Fig. 4. Fig. 5. Fig. 6. An acute angle is that which is less than a right angle (fig. 5). A triangle is a flat surface bounded by three straight lines; when the three sides are equal, the triangle is equilateral; when only two of its sides are equal, isosceles; when none equal, scaline; when one of the angles is a right angle, the triangle is right angled, and then the longest side, or that opposite the right angle is called the hypothenuse. The upper extremity of the triangle is called the apex, the bottom line the base, and the two other including lines the sides. A Quadrilateral figure is a surface bounded by four straight lines. Fig. 7. Fig. 8. Fig. 9. Fig. 10. When the opposite sides are parallel, it is a parallelogram; if its angles are right angles, it is a rectangle (fig. 7); if the sides are also equal, it is a square (fig. 8); if all the sides are equal, but the angles not right angles, it is a rhombus (fig. 9). A trapezio has only two of its sides parallel (fig. 10). A diagonal is a straight line joining two opposite angles of a figure. Plane figures of more than four sides are called polygons. When the sides are equal, they are regular polygons; of which figs. 11-14 are examples, annexed to which are their respective designations. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines, drawn from a certain point within the figure to the circumference, are equal to one another. And this point is called the centre of the circle. The term circle is very generally used for the circumference, and will be found to be employed in this work with this twofold meaning. Fig. 15. Any straight line drawn from the centre and terminating in the circumference is termed a radius; if drawn through the centre, and terminated at each end by the circumference, it is termed a diameter. An arc of a circle is any part of the circumference. A sector of a circle is the space enclosed by two radii and the intercepted arc. When the radii are at right angles, the space is called a quadrant as to one-fourth of a circle. Half a circle is called a semicircle. A chord is a straight line joining the extremities of an arc, as a b. The space cut off by the chord is termed a segment. A tangent to a circle or other curve is a straight line which touches it at only one point, as cd touching the circle at only e. Circles are concentric when described a Fig. 16. from the same centres. Eccentric when described from different centres. Triangular or other figures with a greater number of sides are inscribed in a circle, or circumscribed by it, when the vertex of all its angles are in B the circumference (fig. 17). A circle is inscribed in a straight-sided figure, when it is tangent to all the sides (fig. 18). All regular polygons may be inscribed in circles, and circles may be inscribed in polygons; hence the facility For the measurement of angles, the circumference of a circle is divided G Fig. 19. into 360 equal arcs, called degrees °, which are again subdivided into minutes and seconds"; 60 minutes to a degree, and 60 seconds to a minute, the vertex of the angle being placed at the centre of the circle; the angle is measured by the arc enclosed between the sides. Thus the angle DCB is measured by the arc DB; the line DH, a line drawn from one extremity of the arc perpendicular to the radius passing through the other extremity is called the sine of the angle, GD is the cosine, HB the versed sine, AB the tangent, FE the cotangent, AC the secant, and CE the cosecant. An ellipse is an oval-shaped curve from any point P in which, if straight A F F Fig. 20. lines be drawn to two fixed points FF', their sum will be always the same. FF' are the foci, the line passing through the foci is called the transverse axis, the line CD B perpendicular to the centre of "this line the conjugate axis. A parabola is a curve in which any point P is equally distant from a certain fixed point F and a straight line KK'; thus, PF is always equal to PD-. F is called the focus, and the line KK' the directrix (fig. 21). An hyperbola is a curve from any point P in which, if two straight lines be drawn to two fixed points FF' the foci, their difference shall always be the same (fig. 22). A cycloid is the curve described by a point P in the circumference of a circle which rolls along an extended straight line until it has completed a revolution. If the circle be rolled on the circumference of another circle, the curve then described by the point P is called an epicycloid (fig. 24). Epicycloids are external or internal, according as the rolling or generating circle revolves on the outside or inside of the fundamental circle. The internal epicycloid is sometimes called a hypocycloid. OF SOLIDS. A prism is a solid of which the ends are equal, similar, and parallel straight-sided figures, and of which the other sides are parallelograms. When all the sides are squares, it is called a cube (fig. 25). A pyramid is a solid having a straight-sided base, and triangular sides terminating in one point or vertex (fig. 26). Prisms and pyramids are distinguished as triangular, quadrangular, pentagonal, hexagonal, &c., according as the base has three, four, five, six sides, &c. A sphere or globe (fig. 27), is a solid bounded by a uniformly curved surface, every point of which is equally distant from the centre, a point within the sphere. A line passing through the centre, and terminating both ways at the surface, is a diameter. Fig. 28. Fig. 29. Fig. 80. A cylinder is a round solid of uniform thickness, of which the ends are equal and parallel circles (fig. 28). A cone is a round solid, with a circle for its base, and tapering uniformly to a point at the top (fig. 29). When a solid is cut through transversely by a plane parallel to the base, the part cut off is a segment, and the part remaining is a frustrum of · the solid. The latter term is usually limited to pyramids and cones. Fig. 81. Fig. 82. Fig. 33. Fig. 34. The tetrahedron, bounded by four equilateral triangles (fig. 30). The icosahedron, bounded by twenty equilateral triangles (fig. 34). Regular solids may be circumscribed by spheres, and spheres may be inscribed in regular solids. |