PROBLEMS FOR PRACTICE.

1. Draw five quarter-sections of the rims of wheels, similar in all respects to those in Fig. 42, but placed in the upper right-hand quarter of the

circle.

2. Draw the outlines of a bottomless tub similar to Fig. 43, but of the following dimensions. Extreme upper (front) diameter, twenty-four inches; vertical height, measured on the axis of the tub, twenty-four inches; extreme diameter of bottom of tub, thirty-four inches; thickness of walls, one inch.

Blackboard Exercises.-The foregoing problems.

PAGE SEVEN.-SCALE OF TWO INCHES TO A SPACE. Fig. 47 represents a cubical block, twenty-eight inches square, having semicircular grooves of four inches' radius running diagonally through the centres of its upper, lower, and right and left hand sides, with a circular aperture of four inches' radius running from the front face diagonally through the centre of the block.

Observe that the centres a, b, c, d, e, f, from which the semicircles on the front and on the farther side are described, are situated at the middle point of the edges of the cube, just where they would be placed for marking out the semicircular grooves on the real cube.

Figs. 48 may be supposed to be cut out of a block twenty-eight inches in height (m n or e g); twenty-eight inches in depth (m o or eƒ); and thirty-two inches in width (m e or n g). The figure consists of a frame-work having four corner posts of two by four inches, supporting a half-cylinder above and one below, with their concave surfaces outward, the whole cut out of the block. The thickness of the walls of the half-cylinders is four inches. Their semicircular ends are described from the points a, b, c, and d. The point d, for describing the farther end of the lower half-cylinder, must evidently be at the middle of the farther lower edge of the block, just as c is at the middle of the front lower edge. But c is eight spaces, horizontally, to the left of the corner g; so must d be eight spaces, horizontally, to the left of the corner h. If the pupil will carry out all the measurements, just as they are in the real object, he will find

but little difficulty in the accurate representation of the most intricate plans and patterns.

Fig. 49 represents two cylinders, lying, horizontally and diagonally, side by side on a frame-work of twenty-four by thirty-two inches. As the centres of the farther ends of these cylinders are at the points b and d, the length of their axes, a b and c d, and, consequently, the length of the cylinders, is twenty-four inches. The cylinders are placed, at both extremities, four inches from the ends of the frame. Complete the circle of two-inch radius, of which is the centre, and the circle will strike the frame at the same distance from the farther end that the circle described from c strikes it from the front end.

Fig. 50. This figure consists of two portions, A and B; A being the same as B inverted.

Taking, first, the portion marked 4: the two semicircular curves a and b, of the front face, are described from their common centre, y; while the two marked c and d, of the farther face, are described from their common centre, z. In a similar manner, the two semicircular curves a and b, of the front face of B, are described from w; and the curves c and d, of the farther face, are described from æ.

Fig. 51 consists of a series of the same forms as A, of the preceding figure, here framed together. The pupil should now be able to describe this frame-work in full, and to designate the respective centres from which the several pairs of curves (front curves and rear curves) are described.

Fig. 52 consists of a series of the same forms as B, of Fig. 50, here framed together; or the same as Fig. 51 inverted. The pupil should now be able to describe it in full. In making a drawing of such figures-either in copying, or in drawing from a description-the front portions should be drawn first.

Fig. 53 is a combination of the two figures 51 and 52; requiring merely the placing of the two figures in proper position, and the completing of the vertical corner posts, which are left short in Figs. 51 and 52.

In making a drawing of Fig. 53, the front face should first be drawn, then the right-hand side; then the other parts

should first be traced lightly, and only filled in firmly when the positions of all the parts are distinctly seen.

PROBLEMS FOR PRACTICE.

1. Draw a block the same in size as Fig. 47, but with semicircular grooves of eight inches' radius taken out centrally and diagonally through four of the sides, as in Fig. 47.

2. Draw a figure similar to Fig. 48; but let the height of the front be twenty-eight inches, its width twenty-four inches, and the diagonal depth of the block twenty inches; let the greater curves of the semicircular grooves be struck with a radius of ten inches, and the lesser curves with a radius of eight inches.

3. Draw a solid cylinder of six inches' radius, and twenty-four inches' diagonal length, and let it rest centrally upon a diagonal platform twelve inches wide in front, four inches thick, and twenty-four inches' diagonal length, so that the ends of the cylinder shall be even with the ends of the platform.

Blackboard Exercises.-Figs. 48 and 50, and problems 1 and 3.

PAGE EIGHT.-SCALE OF THREE INCHES TO A SPACE.

Fig. 54 represents a wheel eleven feet and a half in extreme diameter; rim, thirty inches wide and six inches. thick; the cylindrical nave or hub, four feet in diameter and thirty inches in length; and having eight flat spokes or radii, each six inches thick, and the same width as the rim and hub.

The foregoing dimensions being given, we proceed in the following manner to make a drawing of the wheel.

1st. Taking x as the centre of the vertical front face of the wheel, with a radius, 2 a, of five feet nine inches, describe the circle a b c d for the circumference of the wheel, being eleven feet and a half in diameter.

2d. Six inches within the circumference, with the radius xe, describe a circle, which, with the former, will give the thickness of the rim-six inches.

3d. With the radius xf, of two feet, describe a circle for the circumference of the hub, four feet; and with the radius x g, of one foot, describe a circle for the cylindrical opening which is to receive the axle, which is to be two feet in diameter.

4th. Trace lightly the vertical, horizontal, and diagonal

lines through the centre, a, hereby dividing the rim into eight equal parts; and on each side of these several lines lay off a space of three inches, and draw the lines representing the thickness of the spokes, terminating them, in one direction, by the inner circle of the rim, and in the other direction by the outer circle of the hub. This completes the outlines of the front vertical face of the wheel.

5th. As the rim is to be thirty inches wide, and the hub and spokes of the same diagonal extent, we take x y, thirty inches, for the axis, and also for the thickness or width of the wheel. Then y will be the centre of the farther vertical face of the wheel.

6th. From the centre, y, and with the radius y z, of five feet nine inches, describe the outer circumference of the farther face of the wheel; and from the same point y, with a radius six inches less, that is, with a radius, y s, describe a circle (only half of which, hij, will be visible) for the inside circumference of the farther face of the rim; then, with a radius, y r, of two feet, describe a circle (of which k l is a part) for the outer circumference of the farther face of the hub; and finally, with a radius, y p, of one foot, describe a circle (of which m n is the only visible part) for the inner circumference of the farther face of the hub. All the necessary circles will thus be completed-drawn in the very same manner, from the two centres, x, y, of the axis, that they would be drawn from on the real wheel.

7th. The positions and correct width of the front faces of the spokes having been marked out under the fourth division, we have only to draw from their extremities the diagonal lines 1 2,3 4,5 6,7 8, etc., across the inside of the rim and the outside of the hub, and the lines for the farther visible edges of the spokes, and the outlines of the spokes, so far as they are visible, will be completed.

8th. Shade the several parts in such a manner as to make each part distinct from the others, but at the same time paying as much attention as you can, consistent with distinctness, to the laws of light and shade. The light in this drawing, as in most of the preceding, comes diagonally from the upper left-hand corner.

Observe with what accuracy all the parts, in accordance with the scale laid down, and with the principles of representation adopted, maintain their correct measurements, the same as in the real object. Thus all measurements at right angles across the outside or the inside of the rim of the real wheel, and across the outside or the inside of the hub, measure the same, thirty inches. So, also, in the drawing, all the diagonal lines across the rim or the hub-as 1 2,3 4,5 6,7 8, vw, etc.-measure precisely the same, five diagonal spaces, or thirty inches.

Let this wheel be viewed as directed on page 50. The pupil should not only draw this wheel, but also one of a different size, and with the spokes arranged differently.

PROBLEM FOR PRACTICE.

1. Draw a wheel, similar to Fig. 54, of the following dimensions: Extreme diameter, ten feet and a half; rim, thirty-six inches wide and three inches thick; the cylindrical nave, or hub, thirty-six inches in diameter, three inches in thickness, and thirty-six inches in length; and having eight flat spokes or radii, each three inches thick, and the same width as the rim and hub.

Blackboard Exercise.-Let the pupil plan and draw a wheel similar to that described in the problem, and of such size as the board will permit.

PAGE NINE.-SCALE OF THREE INCHES TO A SPACE. Fig. 55 represents a wheel eleven feet and a half in extreme diameter; extreme width of double rim and double hub, four feet; thickness of rim, three inches; extreme diameter of the hub, four feet, and diagonal extent of the two parts, four feet; thickness of rim of the hub, three inches; thickness of the broad paddle-like spokes, three inches, and width four feet.

This figure consists of two similar wheels, the second being placed immediately back of the first, and each having its separate rim and hub, but the two being united in one wheel by the broad paddle-like spokes, which are common to both.

1st. In drawing this figure, draw the front wheel first, in the same manner that Fig. 54 was drawn-o being the centre of the front face of the wheel, and a the centre of the