mains the same; but change the mutual inclination of the sides, and the magnitude of the angle is changed. 7. A quadrilateral in which the sides opposite to each other are parallel, is called a parallelogram. Draw upon your slates the following figures:1. A right angle; an obtuse angle; an acute angle. 2. 6 acute angles; of which each succeeding shall be greater than that which precedes it. 3. 6 obtuse angles; of which each succeeding shall be greater than the one which precedes it. 4. A parallelogram. 5. 2 parallelograms; one with right angles, and the other with oblique angles. 6. Parallelograms in which two of the sides shall be twice, thrice, 4 times longer than the other two. 7. A parallelogram having its sides equal and its angles equal; that is, a square. 8. A parallelogram in which all the angles shall be right angles, but the sides shall not be equal; that is, a rectangle. 9. A parallelogram with equal sides, but unequal angles, that is, a rhombus or lozenge; then a parallelogram with unequal sides and unequal angles; that is, a rhomboid. You will remark that in the rhomboid, as also in the rectangle, the sides opposite to each other are equal. 10. A quadrilateral having two sides parallel to each other, the other two not parallel; that is, a trapezoid; then a quadrilateral having no two sides parallel, that is, a trapezium. 11. A triangle with one right angle; called a right angled triangle, or simply a right triangle. A triangle with one obtuse angle; called an obtuse angled triangle. A triangle with three acute angles; called an acute angled triangle. A triangle with three equal sides; called an equilateral triangle. A triangle with two of its sides equal; called an isosceles triangle. A triangle with no two sides equal; called a scalene triangle. Note.-Other prisms, as the pentagonal, hexagonal, &c., may be examined in a similar manner. 8. A figure bounded by straight lines is called a rectilineal figure or polygon. A polygon is regular if all its sides are equal, and all its angles are equal; otherwise it is irregular. Curvilineal figures are bounded by curved lines; and mixtilineal partly by curved, and partly by straight lines. 9. Draw upon your slates the following figures:— A triangle with equal sides and equal angles, that is, a regular triangle. A regular quadrilateral; a regular 5-sided figure, that is, a pentagon. A regular 6-sided figure, that is, a hexagon. An irregular triangle, quadrilateral, pentagon, &c., to the decagon. THE CYLINDER. 10. Let us now examine the cylinder. A round pillar is a cylinder standing up; a roller' is a cylinder lying down. The cylinder is bounded by 2 plane surfaces, and 1 curved surface. The 2 plane surfaces, called the bases of the cylinder, are parallel; they are of equal size, and each is bounded by a curved line, all parts of which are equally distant from the middle point of the surface. In this upright cylinder the bases are perpendicular to the curved surface, or convex surface of the cylinder, as the whole taken together is called. The convex surface is curved in one direction; in the direction from one base to the other straight lines may be drawn in it. We can suppose a straight line joining the middle of the bases; this is the axis of the cylinder. If we suppose a rectangle to revolve about one of its sides as an axis, the side opposite to this axis will describe the convex surface of the upright cylinder, and the other two sides will describe its bases. We will now examine the oblique cylinder, the fluted cylinder, and the cylinder which has been cut in a direction not parallel to the base. 11. A surface no part of which is plane is called a curved surface. A plane surface bounded by a curved line, all parts of which are equally distant from the middle point, or centre of the plane, is called a circle. The curved line bounding it is called a circumference. The term circle is sometimes improperly used for circumference; but a circle is a surface; a circumference is a line bounding such surface. Any portion of the circumference is called an arc, from a Latin word meaning a bow; a straight line joining the two extremities of an arc is called a chord, from a Latin word meaning a string. A chord divides the circle and its circumference into two parts; each portion of the circle is called a segment; each portion of the circumference is an arc; thus a segment is a surface; an arc is a line. If the chord passes through the centre of the circle it is called a diameter, and it will divide the circle and its circumference into two equal parts, called respectively semi-circles and semi-circumferences. A straight line drawn from the centre to any point in the circumference is called a radius. Two radii drawn in directly opposite directions form a diameter. A straight line which, however far it may be produced in both directions, touches the circumference only at one point, is called a tangent: the point where it touches the circumference is called the point of contact. 12. Draw upon your slates these lines and figures:A circumference; mark the centre of the circle. Draw a radius, a diameter, another chord, and a tangent. Six circumferences of circles having a common centre. Two circumferences intersecting each other. Divide a circle and its circumference into 4 equal parts by diameters. An equilateral triangle in a circle, so that the sides of the triangle shall be chords of the circle. An equilateral triangle about a circle, so that the sides of the triangle shall be tangents to the circle. A square in a circle. A circle about a square. A square about a circle. A circle in a square. A regular pentagon in a circle. A circle about a regular pentagon. A regular pentagon about a circle. A circle in a regular pentagon. Note.-The pyramid, the cone, and the sphere can be examined and treated in a manner similar to that in which we have treated the prisms and the cylinder. 13. Let us now compare together the cube, the triangular prism, and the pentagonal prism. In order to avoid the frequent repetition of the names, we will designate the triangular prism by the letter A, the cube by B, and the pentagonal prism by C. 1. Surfaces. All three bodies are bounded by planes; A by 5; B by 6; and C by 7. The convex surface of each is composed of rectangles; A has 3; B has 6; C has 5. A is likewise bounded by 2 triangles; C by 2 pentagons. 2. Corners or solid angles. A has 6; B has 8; C has 10. 3. Edges or sides. A has 9; B has 12; C has 15. A has 6 sides equal one to the other, and 3 equal one to the other; B has all the sides equal. C has 10 equal one to another, and 5 equal one to another. In 4. Plane angles. A has 9; B has 12; C has 15. A 6 are right angles, and 3 acute angles. In B there are 12 right angles. In C are 10 right angles, and 5 obtuse angles. 5. Line angles. A has 18; B has 24; C has 30 line angles. In A 6 angles are acute, and 12 are right angles; in B all the 24 angles are right angles; in C 20 are right angles, and 10 are obtuse angles. At each corner of A are 2 right and 1 acute angles; at each corner of B are 3 right angles; at each corner of C are 2 right and 1 obtuse angles: thus we have 2 right angles at each corner of each of these bodies. |