DRAWING-BOOK No. II.

CABINET PERSPECTIVE-PLANE SOLIDS.

THE Cabinet Perspective presented in this series is a method of representing solids, both plane and curvilinear, in such a manner that the drawings shall give, by the aid of the ruled paper, the correct measurements of the objects represented. The ruling on the paper is adapted to any scale of measurement; but, for the purposes of the present illustration, let it be supposed that the vertical and horizontal lines on the paper are respectively one inch apart.

In all drawings in what is called Diagonal Cabinet Perspective,* objects are supposed to be viewed in a manner similar to that in which the two cubest in Fig. 1, on page 1 of Drawing-Book No. II., are represented. Taking the cube at B for illustration, this may be supposed to be a cube six inches square, the front face of which is in a vertical position. The spectator is supposed to view this cube from such a point, above and at the right of the cube, that he may see just as much of the upper side of the cube as of the right-hand side; so that the apparent width, 10 9 of the upper face, or 12 0 of the side face, shall measure, in the directions indicated, one half the width of the front face; and so that the diagonal corner lines, 1 2,3 4, and 5 6, will seem to rise diagonally at an angle of forty-five degrees; while the distance from which the view is taken is supposed

*There is a beautiful practical application of Cabinet Perspective, called Semi-diagonal Cabinet Perspective, which will be found illustrated on pages 9-11 of Drawing-Book No. IV. See page 171.

† A cube is a regular solid body, having six equal square sides.

to be so great that these lines will appear to be, as they are here drawn, parallel.

The front face of the cube is drawn in its real proportions as a square, and as though it were seen in a vertical plane directly fronting the spectator. According to the scale supposed to be adopted in the ruling, the front of the cube measures six inches to a side. The farther face of the cube, being also vertical and parallel with the front face, and therefore in a plane also directly fronting the spectator, would also be drawn as a square if it could all be seen, measuring six inches to a side. Hence each of the lines 2 4 and 4 6 measure six inches, the same as 1 3 and 3 5.

But each of the diagonal lines 1 2, 3 4, and 5 6, being corner lines of the cube, must also represent a measure of six inches; and as each of these lines extends over three diagonals of the small squares, it follows that what we call one diagonal space measures twice as much as a vertical or a horizontal space, whenever this diagonal space is applied to the measurement of a line representing a horizontal line. We may, therefore, adopt the following rule for the measurement (and also for the drawing) of all objects in diagonal cabinet perspective.

ELEMENTARY RULE.

Drawings of surfaces that are supposed to be in a vertical plane fronting the spectator are measurable, in any direction, according to the scale adopted for the vertical and horizontal spaces of the ruling; while each DIAGONAL space that measures a line in a horizontal and diagonal position is to be taken as TWICE the measure of a space of the other kind.

Applying this rule to the cube at B, Fig. 1, we find that all the horizontal lines, and also all the vertical lines that cross the front face of the cube, measure each six inches in length, because each extends over just six of the ruled spaces, and all are in a vertical plane fronting the spectator. For a similar reason each one of the vertical lines on the right-hand side of the cube, and each one of the horizontal lines on the top of the cube, measures six inches, because each may be supposed to be in a plane like that which

forms the front face of the cube-directly fronting the spectator. But such lines as 1 2,3 4, 5 6,8 9, 10 11, and 12 13, being seen obliquely, can not be in any plane fronting the spectator; and as they lie in a diagonal direction, and represent horizontal lines, they are measurable by the principle adopted in the latter part of the rule. Hence the line ab measures five inches; c d four inches; g 13 four inches; m n six inches: but 8 9 measures six inches; 10 11 meassures four inches; 12 13 measures four inches, etc. The cube at A measures two inches on each of its corners.

Fig. 2. Applying the scale of measurement which we have adopted to the representation of the cube at D, Fig. 2, we find that the front face of the cube is a square of four inches to a side; and that the diagonal horizontal distance 1 2, or 34, or 5 6, also measures four inches. Also, if we draw intermediate lines between the ruled lines on the paper, on the upper face and right-hand face of the cube, so as to give one-inch diagonal measures, then sixteen one-inch squares may be counted on each of the three visible faces or sides of the cube. We thus have, according to the scale of one inch to a space horizontally or vertically between the lines, and two inches for a diagonal space, the exact measurement of the three visible sides of the cube.

To find the contents of a cube:

RULE.-Multiply the length of a side of the cube by itself, and that product again by a side, and this last product will give the contents required. (See Rule I., page 53, for the measurement of surfaces.)

Thus, in the cube at D, Fig. 2, if we multiply the length (1 2) of one side, which is four inches, by the length (2 4) of another side, which is also four inches, we get the product 16, which is the number of solid cubic inches contained in the upper tier of the cube-as may also be seen by counting them; and as there are four of these tiers, we multiply the 16 by 4, and get 64, the number of cubic inches in the four tiers, or in the whole cube. Or, 4×4×4=64 cubic inches.

One cubic inch is represented at C, which, according to

the scale we have adopted for page 1, measures one inch on each of its sides; and at E are sixteen cubic inches, equal to the upper tier in D.

In straight-line drawings in cabinet perspective, the ruler may be used wherever its aid will give additional accuracy to the drawing.

The contents of the cube B, Fig. 1, which measures six inches to a side, are found by the rule to be as follows. Ans. 6×6×6=216 cubic inches. The drawing at E, Fig. 2, is an example of a parallelopi ped; a figure which is defined as being a solid whose faces are six rectangles,* the opposite faces being parallel, and equal to each other. The drawing at Falso represents a parallelopiped. All squares are parallelopipeds; but all parallelopipeds are not squares.

To find the contents of a rectangular (right-angled) parallelopiped:

RULE.-Multiply the length by the breadth, and that product by the depth, and this last product will give the contents required.

The height of the rectangular solid at E, Fig. 2, is one inch; the breadth in one direction (1 2) is four inches; and the breadth in the other direction (3 4) is also four inches. What are the contents? Ans. 1×4×4=16 cubic inches, as may be verified by counting the small one-inch cubes which it contains.

The length (or height) of the solid at Fis three inches; the breadth or width is five inches; and the diagonal depth is two inches. What are the contents?

Ans. 3 × 5 × 2=30 cubic inches. The two rules just given may be combined in one, as follows:

To find the contents of any solid rectangular figure:

RULE.-Multiply the three dimensions together, and their product will be the contents required.

Fig. 3. At G, H, I, and J are represented four parallelo

* In a strict definition the sides need not, necessarily, be rectangular (right-angled); but it is better, at present, for the pupil in drawing to consider all parallelopipeds as of the rectangular kind.

pipeds, all of the same size; G being viewed in a vertical position, H horizontally, and I and J horizontally and diagonally. They may be considered pieces of timber, each two inches square at the ends, and twelve inches long. What are the solid contents of each ? Ans. 48 cubic inches.

Observe that, according to the scale of measurement already explained, these pieces measure precisely the same in these three different positions.

Fig. 4. According to the definition of a parallelopiped, this figure, also, is one of that kind. What are its meas

urements, and its contents?

Observe that the right-hand sides of the foregoing figures are represented as shaded with a deep tint of India ink, the front with a lighter tint, and the top of Fig. 4 with the running dotted shading. The kinds of shading used in Fig. 4 are well adapted to all plane solids, as the object of shading, in cabinet perspective, is to render the several surfaces as marked and distinct as possible, one from another.

Fig. 5 is a square frame composed of four pieces, each two inches square; the two diagonal side-pieces each twelve inches long, and the other two each eight inches long. What is the size of the square which they inclose?

Fig. 6. Let the pupil describe Fig. 6-that is, tell how many pieces compose the figure, their size, position, etc.

Fig. 7 is drawn, first, in the same manner as G of Fig. 3; it is then divided so as to represent cubical blocks, each two inches square, placed one above another. Three of these blocks are represented as shaded with the hatching described in Lesson IX. of Drawing - Book No. I., after first tinting the surface with India ink.

Fig. 8 is composed of two vertical blocks, each two inches. square and five inches long, resting upon the ends of a piece one inch by two inches, and twelve inches long; the latter being viewed diagonally.

Fig. 9 is the same as No. 8 inverted, and still viewed diagonally.

Fig. 10 is also the same as Fig. 8; but it is here viewed horizontally. Thus the same figure may be represented in several different positions, so as to bring each side into full view.