DRAWING THE OUTLINES OF THE SOLID BODIES. 14. We will now endeavor to represent upon a plane surface the 5 faces of the triangular prism in connexion. This body is bounded by 2 equilateral triangles opposite one to the other, and 3 rectangles in contact one with another; and our drawing, or diagram, must conform to this. We must therefore construct 3 rectangles in contact, and the 2 triangles, one at each end of one of the rectangles. (Fig. 1.) In a similar manner may be represented the quadrangular, pentagonal, and hexagonal prism; the cube; the cylinder; the cone; the triangular, quadrangular, pentagonal, and hexagonal pyramid. (Figures 2 to 11.) Note. It would be useful to have these representations of the solid bodies drawn on pasteboard, the figure cut out all round, and incisions made in the lines where the several parts of the bodies unite, so that the pasteboard may bend and form the bodies drawn upon it. By means of these the truth of the representations will at once be perceived. THE REGULAR SOLIDS. 15. A regular solid is one, all the faces of which are equal regular polygons; that is, in which all the line, plane, and solid angles, all the faces, and all the sides, are respectively equal. There are five regular solids, the names of which are formed from the Greek word signifying seat, or face, combined with the Greek numeral denoting the number of seats or faces in the solid. 1. The tetraedron, which is bounded by 4 equilateral triangles. 2. The octaedron, bounded by 8 equilateral triangles. 3. The icosaedron, bounded by 20 equilateral triangles. 4. The hexaedron, or cube, bounded by 6 squares. 5. The dodecaedron, bounded by 12 regular pentagons. The name polyedron is used to denote any solid bounded by planes, whether it be regular or irregular. It is derived from the Greek, and signifies many-faced. Exercises, similar to those which have preceded, may be performed with the regular solids. It will be very useful to have models of these solids, made either of wood or pasteboard. The scholars may make these models for themselves, of clay, turnips, potatoes, &c. It will be very useful for them to draw upon paper, or some plane surface, a skeleton, or representation of the faces of each, as they are shown in figures 5, 77, 78, 79, 80. EXERCISES ON FIGURES. 16. The teacher may dictate to the scholars a figure, that is, he may tell them what lines to draw upon their slates; or he may himself draw a figure upon the board, and desire the scholars to tell, or draw, all that they remark in the figure. A few examples will suffice to show the mode of proceeding. Write down all that you remark in this figure, (having drawn figure 12.) The scholars will find as follows. Figures. 1 square; 4 triangles, -each composed of 2 triangles; 4 smaller triangles; 4 pentagons. Lines. 6 lines,-4 which bound the square, and 2 inside the square; 3 lines meet at each corner; the two interior lines cut each other at one point. Angles. 4 angles in the square; 4 at the point in the middle; the 4 angles of the square are divided into 8 parts; in the 4 large triangles are 12 angles; and in the 4 small triangles there are likewise 12 angles; 4 right angles in the square; 4 right angles at the middle point; 8 acute angles in the triangles. 17. The bounding lines of the square, or of any polygon, taken together, form the perimeter of such polygon. The straight lines within the square are called diagonals; thus a diagonal is a straight line joining the vertices of any two angles of a polygon, which are not already connected by one of the bounding lines. 18. Tell me all that you remark in this figure. (Fig. 13.) Figures. Rectilineal. The same as in fig. 12. Curvilineal. 1 circle. Mixtilineal. 4 large and 4 small two-angled figures; 4 large and 4 small triangles. Lines. Straight. The same as in fig. 12. ferences; 4 quarter-circumferences; 4 three-quarter-circumferences. Angles. Rectilineal. The same as fig. 12. Mixtilineal. 8 mixtilineal acute angles; 8 mixtilineal right angles: 8 mixtilineal obtuse angles. 19. We will now draw a figure from dictation; each one upon his slate. A straight line; bisect it, that is, divide it into 2 equal parts; at the division point draw a line perpendicular to the first line, and of the same length; bisect the perpendicular; connect the end and middle points of the perpendicular by straight lines with the end and middle points of the first line. Now write down all that has been done; mention the figures that are formed; and whatever you remark in those figures. 20. In the preceding exercises we have become acquainted with : 1. A Solid, which has extension in length, breadth, and thickness. 2. A Surface, which is the boundary of a solid, having extension in length and breadth only. 3. A Line, which is the boundary of a surface, and has extension only in length. 4. A Point, which is the extremity of a line, and has only a position; it has no extension, either in length, breadth, or thickness. The science which treats of the measure of extension is called Geometry, from two Greek words which signify to measure land; thus denoting the purpose to which the science was first applied. PART SECOND. RECKONING AND CONSTRUCTION. I. POINTS. 21. LET us consider how often 2, 3, 4, or more points can be placed in a different order of succession. We will first take 2 points, which we will designate by the letters a and b. We may have a on the left, b on the right; a on the right, b on the left. Thus we have 2 ways, ab, ba, that is, 1 × 2. 3 points, a, b, c. Two points give 2 ways, ab, ba. Now take c with ab. It may occupy the 3d, 2d, or 1st place. This gives abc, acb, cab, 3 ways. Now take c with ba, and we shall again have 3 ways, bac, bca, cba, in all 6 ways, that is, 1 × 2 × 3. 4 points, a, b, c, d. We can take the six preceding ways, and add the 4th point d to each of the 6. The 1st was abc; add d; it may occupy either the 4th, 3d, 2d, or 1st place; thus, abcd, abdc, adbc, dabc, |