APPENDIX.

ISOMETRICAL DRAWING.

I. ELEMENTARY PRINCIPLES.

ISOMETRICAL DRAWING, or Isometrical Perspective, is based upon the following principles: If a cubical block, as shown in Fig. 1, Plate I., and as seen shaded in Fig. 2, be supposed to be viewed from an infinite distance, and from such a position that the line of vision shall pass through the upper and nearer corner, 1, and also through the lower farther corner, the three visible faces, A, B, C, of the cube will appear to be equal in measure, the one to the other. Any boundary-line of the upper surface, A, will measure the same as any boundary-line of the face B, or of the face C. Thus the line 14 will measure the same as the line 1 6, or 4 5, or 6 5, or 1 2, or 2 3, etc. And any measure taken on any one line will give the same relative distance when applied to any other line. Hence the appropriateness of the term Isometrical, which is formed from two Greek words signifying equal in measurement.

The isometrical cube is based upon the geometrical principle for inscribing a hexagon in a circle. Thus, to inscribe a hexagon in the circle, Fig. 1, take the radius, 1 4, and, beginning at 2, apply it six times to the circumference, and it will give the points 2, 3, 4, 5, 6, 7. Join these points by straight lines, and we shall have the six equal sides of a regular hexagon. Connect the alternate corners of the hexagon with the central point, 1, and retain the circumference of the circle, and we shall have the isometrical cube inscribed in a circle.

In the isometrical cube, which is supposed to stand upon a horizontal surface, while the spectator looks down upon it diagonally, each of the lines 2 3 and 2 7 forms an angle of sixty degrees with the vertical line 2 1, and an angle of thirty degrees with the horizontal line 8 9. It will be observed, also, that the lines 1 4 and 6 5 are parallel to 2 3; 16 and 4 5 parallel to 27; 34 and 7 6 parallel to 2 1. In the isometrical cube, therefore, and in all isometrical drawing, there are only three kinds of true isometrical straight lines-vertical lines, and the two kinds of diagonals as seen in Fig. 1. But unless the diagonal lines form exact angles of sixty degrees with the vertical lines, the drawings made on them will be distorted; and as these lines can not be made with sufficient accuracy with the pen or pencil, we have had them accurately engraved, and printed in red ink on drawing-paper. By the aid of such paper all difficulty in making accurate isometrical drawings is now removed, as the ruling is a perfect guide for the direction of all the diagonal lines; and the vertical lines, as will be seen, follow the intersections of the diagonals.

Let it be observed, also, that the diagonal distance from point to point in the intersections of the diagonal lines is precisely the same as the vertical distance between their intersections. Thus, in Fig. 1, the five diagonal spaces from 1 to 4, or 1 to 6, or 2 to 7, etc., measure the same as the five vertical spaces from 1 to 2, or 4 to 3, or 6 to 7, etc. Moreover, the Isometrical Drawing-Paper is so ruled as to correspond, in measure, to the ruling of the Cabinet DrawingPaper-what is called a space in the one corresponding to a space in the other. From the foregoing explanation, the pupil who is familiar with the principles of cabinet perspective will have little difficulty in making every variety of plane isometrical drawings.

II. FIGURES HAVING PLANE ANGLES.

PLATE I.-SCALE OF TWO INCHES TO A SPACE.

According to the scale here adopted, Fig. 1-represents the outlines of a cube of ten inches to a side, and Fig. 2 is the same, shaded. The student will notice the difference between the mode of measurement here adopted and that used in cabinet perspective. In the latter, also, one face of the cube-the front vertical face-would be drawn in its natural proportions, as a perfect square.

Fig. 3 represents a cube of ten inches to a side, having rectangular pieces six inches square and two inches in thickness cut from the centres of its three visible faces. Let the pupil compare this drawing with that of Fig. 12, page 1, of Drawing-Book No. II. The cube shows to excellent advantage in isometrical drawing.

Fig. 4 represents an inverted frame sixteen inches square, with corner posts two inches square and six inches in length.

Fig. 5 is the same as the English cross bond shown in Fig. 38, page 4, of Drawing-Book No. II. The two figures illustrate, very happily, the two methods of representation -cabinet and isometrical. The scale adopted being the same in both cases, the bricks measure the same in both.

Fig. 6 is the same as the upper part of Fig. 88, page 11, of Drawing-Book No. II. A figure of this kind, evidently, does not show to so good advantage in isometrical as in cabinet drawing. The former is best adapted to the representation of objects whose side views are nearly equal in proportion.

PROBLEMS FOR PRACTICE.

We would now recommend the pupil to draw, in isometrical perspective, all the figures given on the first five pages of Drawing-Book No. II. Let him take the measures as there indicated by the scale, but let him remember that a diagonal space is there to be taken as twice the length of a vertical or horizontal space, while in isometrical drawing a diagonal space and a vertical space measure the same, and are to be considered of equal length.

The pupil would do well to draw all the problems, also, connected with these first five pages.

PLATE II.-SCALE OF ONE FOOT TO A SPACE.

We have here adopted a scale of one foot to a space, although any scale whatever, that is most convenient, may be used.

Fig. 7 is intended to represent the upper part of a pillar five feet square, around which a moulding of one foot projection and one foot in height is to be placed, even with the top. The shaded portion shows the attachment of the moulding to the pillar.

Fig. 8 shows the moulding as attached, and concealing from view a portion of the pillar down to the line 8 9 10. Hence the following rule:

RULE. Any horizontal rectangular moulding attached to a vertical surface obstructs the view of that surface below the moulding to an extent equal to the extent of the projection of the moulding.

Fig. 9. The dotted outline represents a cubical block four feet square, while the shaded portion shows a wedge cut from it. The sides of the wedge bevel off equally from the sharp edge 1 2, inasmuch as the lines 14 and 1 5 intersect the base line 45 at equal distances from the point 3.

Fig. 10. The pillar in this case is of the same size as that seen in Fig. 8; but in Fig. 10 the moulding is cut up into three cubical blocks on a side, each one foot square.

Fig. 11 shows how triangular blocks attached to the top of a column may be represented. The dotted continuations of the lines of the farther two blocks show the concealed points on the column toward which the lines are to be drawn.

Fig. 12 represents a truncated pyramid, the base of which is surrounded by rectangular mouldings. Observe that the side lines of the pyramid are drawn toward the point x.

Fig. 13. Very tall four-sided pillars, gradually tapering, and having a flat pyramid at the summit, as at A and B, are called obelisks. Observe that the apex, in these two obelisks, is in the central vertical line of the pyramid.

Fig. 14. This pyramid has a rectangular section, one foot in depth, cut from each of the two visible sides of the base,

and triangular sections cut through the pyramidal portion, so that all except the four edges of the pyramid, and one foot in thickness of its base proper, are cut away. The farther edge of the pyramid is concealed by the front edge.

PLATE III.-SCALE OF TWO FEET TO A SPACE.

Although any object drawn isometrically is supposed to be viewed in the direction of the diagonal of a cube, yet we may view any one face of a cube, or any one side of any rectangular object, from four different positions, and at the same time view it in the direction of some one of the diagonals of a cube. Thus:

Fig. 15 represents a block viewed in the direction of the diagonal that passes through the corner 1. We here see the top, front, and right side.

We here

Fig. 16 represents the same block viewed in the direction of the diagonal that passes through the corner 2. see the top, front, and left side.

Fig. 17 represents the same block viewed in the direction of the diagonal that passes through the corner 4. We here see the bottom, front, and right side.

Fig. 18 represents the same block viewed in the direction of the diagonal that passes through the corner 3. We here see the bottom, front, and left side.

Figs. 15 and 16 are viewed from above, and 17 and 18 from below. These are similar to the different views of objects in cabinet drawing, as represented on page 1 of Drawing-Book No. IV.

Fig. 19 is the same as Fig. 56, of page 7, in Drawing-Book No. II., although the designated scales are different. By adopting the same scale, the figures will measure alike.

Fig. 20 represents two flights of steps, ascending in different directions, and leading to a platform seven feet in height. As each step rises half a space—that is, one foot, seven steps are required to reach the platform.

Fig. 21. The upper roof, A, of this structure is evidently horizontal. The second portion, B, declines downward from the horizontal, as represented by the extent to which the line 1 3 diverges downward from the diagonal horizon

I