In these several figures observe the effect of the shading, which is rapidly executed with different tints of India ink, except the upper surfaces, which have the running dot-line shading. PROBLEMS FOR PRACTICE. 1. Draw the representation of a square frame similar to Fig. 5, of the same outside and inside measure as Fig. 5, but composed of pieces only one inch thick instead of two inches; and all of them two inches wide. 2. Draw a figure similar to Fig. 6, but let the upper pieces be only one inch thick (or high), and let them project each one inch at both ends beyond the lower pieces. 3. Draw a figure similar to Fig. 7 in position, but four inches wide and two inches thick-the wide side being in front. 4. Draw a figure similar in position to Fig. 8, but composed of three pieces, each twelve inches long, four inches wide, and one inch thick. 5. Draw Fig. 9 with the side in a plane fronting the spectator. Free-hand Blackboard Exercises.-Figs. 5 to 10 inclusiveletting a space on the blackboard represent its true measure, two inches. In this case only one half of a diagonal measure will be required for a distance of two inches. In order to render the several surfaces distinct one from another, the dark shades may be represented by heavy vertical chalk lines, and the lighter shades by very light vertical lines. Or, where colored chalk crayons are accessible, the dark shades may be represented by blue lines. Fig. 11, in its complete outline, is a cube eight inches. square, having a small cube two inches square cut from each of its four upper corners. See the form of the entire cube, as represented by the dotted lines. What were the contents of the entire cube? What the contents after the four corners were taken out. Fig. 12 is a cube twelve inches square, having a piece eight inches square and two inches thick taken from the centre of each of its three visible sides. What were the contents of the entire cube before the three pieces were taken out? What were the contents after these pieces were taken out? Let the pupil be careful, in drawing the figure, that his measures shall be correct. Thus, the depth of the recess in each side must measure two inches. Thus c d and m n are measures of two inches each; and a b, being one diagonal, is also a measure of two inches. Fig. 13 represents a rectangular frame measuring ten by fourteen inches, and composed of pieces two inches square framed into posts two inches square and ten inches long. To get the full effect of the figures on this page, view them from a point at the right, and above them, through the opening formed by the partially closed hand. The pupil should be accustomed to view his drawings in the same manner. PROBLEMS FOR PRACTICE. 1. Draw a figure similar to Fig. 11, but twelve inches square, having a block four inches square cut from each of its four upper corners. What would be the contents of the entire cube? What the contents after the four corners were taken out? 2. Draw a figure similar to Fig. 12, but fourteen inches square, and showing a rectangular piece ten inches square and four inches in thickness cut from the centre of each of its three visible sides. What would be the contents of the entire cube? What the contents after the three rectangular pieces were taken out. 3. Draw a figure similar in all respects to Fig. 13, except that the four horizontal pieces of the frame-work are to be only one inch in vertical thickness. Free-hand Blackboard Exercises.-Figs. 11, 12, and 13; also the accompanying problems. PAGE TWO.-SCALE OF ONE INCH TO A SPACE. Fig. 14 represents a cube eight inches square; and Fig. 15 represents a box formed of one-inch stuff, open at the top, and of a size that will just receive the cube; so that the latter, when placed within the box, shall fill it even with the top. At A is the cover that will just fit the top of the box. What is the size of the cover? Of the side B of the box? Of the sides CC? Of the bottom of the box? What are the outside measures of the box when the cover is on it? Fig. 16 represents a series of five blocks placed one upon another, and rising in the form of stairs. The upright piece in a stair (as a) is called the riser, and the part on which the foot is placed (as b) is called the tread. In Fig. 16 the stairs are so placed that the risers front the spectator; but in Fig. 17 the side of the stairway fronts the spectator, and the riser is viewed diagonally. Fig. 17 measures the same as Fig. 16, with the exception that in Fig. 17 the lower block is omitted. If these two figures are supposed to be drawn to a scale of four inches to a space, what will be the height of each riser, and the width of each tread? What the width of the stairway? Fig. 18 represents a cabinet frame-work formed of pieces. two inches square at the ends; the whole frame measuring ten inches by twenty-six. Here the longest side fronts the spectator; but in Fig. 19 the same frame-work is represented with the end fronting the spectator. Observe that the end below c d of Fig. 18, which is there seen diagonally, is not seen at all in Fig. 19; and that the end below a b of Fig. 19 is not seen at all in Fig. 18. Let the pupil describe the several pieces of which the frame-work is composed. Fig. 20 represents a cross made of pieces measuring, at the ends, two by three inches. Fig. 21 represents a cabinet square made of one-by-twoinch stuff, and placed horizontally; but, as the spectator is supposed to be above and to the right of it, one of the arms seems to rise at an angle of forty-five degrees-being in the diagonal direction of the small ruled squares. Fig. 22 represents the same cabinet square in a vertical position, and so placed that the wide side shall front the spectator. The former, as placed, is a measure for horizontal and diagonal distances; the latter, for horizontal and vertical distances. Fig. 23 is an upright frame-work resting on four blocks, each two inches square; having only two of its sides and the top and bottom inclosed, and containing three shelves in addition to the top and bottom. Let the pupil describe this frame-work more fully; and tell (or write out) its measures in all its parts-distances apart of the shelves, etc. The whole on the supposition that it is drawn to the scale of one inch to the space. The shading on this page is supposed to be done, first, with India ink; some of the sides are then shaded with lines by the pencil, which, when not too heavy, give additional life and spirit to the drawing. PROBLEMS FOR PRACTICE. As Fig. 14 represents a cube eight inches square, what are its contents in cubic inches? Ans. 8x8x8=512 cubic inches. As it takes 231 cubic inches to make a gallon, how many gallons of water would the box, Fig. 15, contain? Ans. 2 gallons and 50 cubic inches. 1. Draw a box, similar to Fig. 15, that would contain a cube twelve inches square; and draw the cover separately. How many gallons of water would such a box contain? 2. Draw a rectangular box open at the top, the bottom of which shall measure, on the inside, ten by twelve inches, and let the box be ten inches deep. How many gallons of water will it contain? 3. Draw a flight of stairs, similar to Fig. 17, to a scale of six inches to a space, and having the risers twelve inches high, and the tread eighteen inches wide. 4. Draw a frame similar to Fig. 18, but formed of stuff one by two inches -the one-inch being the height. Free-hand Blackboard Exercises. Figs. 17, 18, and 20, and problems 2 and 3. PAGE THREE.-SCALE OF THREE INCHES TO A SPACE. Fig. 24. At A is represented a piece of timber six by twelve inches at the end, and four feet long, broadest side down, having a piece cut from the centre of the upper side one foot square and three inches deep, to receive crosswise another timber cut in like manner. At B the two pieces of timber are united at right angles to one another. The remaining figures on this page are examples of what carpenters call scarfing, which means the uniting of two pieces of timber longitudinally by a scarf-joint-in common language often called splicing; but the latter term is more properly applied to an overlapping joint that is not notched. Fig. 25. At C and D are two pieces of timber prepared for being joined lengthwise by a square scarf-joint. After placing the timbers in the right position, they are pinned or bolted together. Let the pupil describe the two pieces. Fig. 26. At E is the upper piece of timber, and at F the lower, to be united by a scarf-joint, which is formed of the square scarf-joint combined with the splice-joint. Observe that the line of union, 2 4, in both E and F, is in the direction of two-space diagonals; and as the two lines are of the same length, and the two timbers lie in the same relative positions, the cut sections must correspond with one another. Moreover, E and F are redrawn at G and H; and there it is seen that the diagonal line of union, 24, is in both cases the diagonal of the same rectangle 1 2, 34. Fig. 27 shows two pieces of timber, I and J, prepared for being joined by the ordinary splice-joint. The two timbers are to be firmly bolted together. But this is by no means so firm a mode of union as is shown in Figs. 25 and 26. Fig. 28 shows a still firmer mode of the scarf-joint than Figs. 25 and 26. Observe that the lines of union in K measure precisely the same, and are in precisely the same positions, as in L. Thus the lines 1 2 and 3 4 are the same in length and position as the lines 5 6 and 7 8, both being two-space diagonals. Fig. 29 shows, perhaps, the firmest of all modes of the scarf-joint, especially for heavy timbers. N is the upper piece; and the two must be united by placing M and N side by side, and driving them together. Even then, without bolting, they can not be drawn apart lengthwise; nor, if the joints be good, can they be easily sprung in any direction. At X the two pieces are shown as they appear when united. Fig. 30 represents the same pieces that are shown in Fig. 29; but here drawn in a different position, being placed sidewise toward the spectator, instead of end wise toward him as in Fig. 29. Observe that the measures, according to the scale, are the same in the one case as in the other. That mode of representation which will give the best view of the object should in all cases be adopted; and in some cases both modes should be used, as the one will often give views of parts that are not shown in the other. Fig. 31 shows a still different mode of scarfing, and one that is very firm, and much more easily executed than Fig. 30. When the pieces are firmly bolted together, the inter |