tal line 12. If the portion B were horizontal, and of equal width on both sides, the line 6 3 would extend to 7, and the line 39 would coincide with 7 8.

Again: if the portion C were vertical, the line 3 4 would coincide in appearance with the line 3 5. The other portions of the structure require no explanation.

PROBLEMS FOR PRACTICE.

The pupil ought now to find no difficulty in changing all the figures from page 6 to page 11 inclusive, and Figs. 99, 100, 101, and 102, of page 12, of Drawing-Book No. II., into isometrical drawings. If he think this would require too much labor, he would do well to work out the problems, at least, isometrically.

III. THE DRAWING OF ISOMETRICAL ANGLES. The rectangular ruling on the upper part of Plate IV. is designed to correspond precisely in the measure of its spaces—that is, in the distance from line to line, measured on the lines-to the isometrical ruling on the lower part of the plate. So, also, the ruling on the "Isometrical DrawingPaper" corresponds, in like manner, to the ruling on the "Cabinet Drawing - Paper." On this basis we have constructed a "Scale of Angles," which is applicable alike to the drawing of angles on both isometrical and rectangular bases. For inasmuch as any one of the four angles of a rectangular square may be divided into ninety equal degrees, so also may any one of the four angles of a corresponding isometrical square be divided into equivalent isometrical degrees, isometrically representing the angles of the rectangular square. Thus:

Fig. 22. Scale of one foot to a space. In the geometrical square A B C D, the quarter-circle B D is divided, by the full lines which diverge from A, into nine equal parts, each part representing ten degrees at the corner A. The division is best made by the compasses, in the following manner:

From D, with the distance D A, cut the curve B D at s; and from B, with the same distance, cut the curve at t.

The curve B D will thus be divided into three equal parts, representing angles of thirty degrees each at the point A. Next divide each of these parts, by the compasses, into three equal portions, and the entire curve will then be divided into nine equal parts, of ten degrees each. Through these points of division draw lines from A, and extend them to the sides B C and D C of the square. Draw a dotted line from A to C, and the angle D A C will be half a right angle-that is, an angle of forty-five degrees, while each of the angles D A 10, 10 A 20, 20 A 30, etc., will be an angle of ten degrees.

Within the larger square, A B C D, we may count twenty-five different squares, each having one of its angles at A; and on the two sides of each of these squares, opposite A, we have the same degrees marked off, by the lines diverging from A, that we have on the sides B C and D C of the larger square. Thus the measure 8 p, on the side of a square of eight spaces, measures an angle of twenty degrees at A, as truly as the measure D 20 measures the same angle; and 8 measures an angle of forty-five degrees, just as effectually as D C measures the same angle.

Now, inasmuch as any one of the twenty-six rectangular squares that may here be designated exactly measures an isometric square of the same number of spaces to a side, the measures of angles on any one of these rectangular squares may be used to lay off like angles on a corresponding isometric square. Thus:

Fig. 23. It is required to lay off, from the point 1 in the line 1 2, an angle of ten degrees. As the lines 1 2 and 2 3 are two sides of an eight-space isometric square, they correspond to the two lines A 8 and 8 r (in Fig. 22), two sides of an eight-space rectangular square, and measure the same. From the point 8, on the line A D, take the distance 8 a, and apply it to the isometric square on the line from 2 to 3, and mark the point 5. A line drawn from 5 to 1 will then correspond to the line a A; and the isometric angle 5 12 will correspond to the angle a A 8, and will represent an angle of ten degrees.

If from the point 4, in the line 4 3, of the isometric square

of eight spaces, we would lay off an angle of twenty degrees, lay off 3 7 equal to 8 p; draw a line from 7 to 4; and the angle 3 4 7 will be an isometric angle of twenty degrees, the same as 8 A p is an angle of twenty degrees.

To lay off an angle of twenty degrees from the point b in the line b d, make d c equal to 8 p, and connect c b. The angle cbd will then be an angle of twenty degrees.

To lay off an angle of forty degrees at the point g in the line gh, form an isometric square, as g h k n, of five spaces, and from h lay off h i equal to 5 m of the five-space rectangular square, and connect g i. Then h g i will be an isometric angle of forty degrees, the same as the angle 5 A m is an angle of forty degrees.

The angles laid off in Figures 24 and 25 may now be easily drawn. It is not necessary, in any case, to lay off a full isometric square to correspond to the rectangular square. It is sufficient to have one side of the isometric square, and enough of the other side to receive the measure from the rectangular square.

If, in Fig. 19, Plate III., it be required to make 2 1 3 a certain angle, the angle may be laid off in the manner just illustrated. The same with any other angle which it may be required to draw on any isometrical square. So also, in Fig. 21, if it be known what angle the line 3 1 forms, in the real object, with the horizontal line 1 2, or 1 7, the angle may be laid off from the scale, by considering that 12 or 1 7 corresponds to a portion of the line AC of the scale. The angle 7 1 10 is then an isometric angle of fortyfive degrees. So also may the angle 4 3 5, if it be known, be laid off from the scale, inasmuch as the lines 4 3 and 5 3 appear just as they would if they were in a vertical plane that coincided with 1 2.*

*Note.-The scale shown in Fig. 22 may be applied to the drawing of definite angles in cabinet perspective, when the measures of angles can be taken on that edge of a cabinet square which measures the same number of spaces as the edge of a corresponding rectangular square.

Thus in the cabinet cube B, Fig. 1, page 1, of Drawing-Book No. II., which is a cube of six spaces (six inches) to a side, angles at 5 or 3, up to forty-five degrees, may be taken from the scale and laid off on the side 6 4;

IV. THE ISOMETRIC ELLIPSE AND ITS APPLICATIONS. The Isometric Ellipse is the ellipse which is drawn within an isometric square, touching the middle points of its sides, as the three ellipses in Fig. 26, Plate V. The isometric ellipse represents a circle viewed in the position of a side of an isometric cube.*

Fig. 26. Plate V.-Here is represented a cube which measures ten spaces to a side, and on each of its three visible faces is an isometric ellipse which represents a circle drawn touching the middle of the sides of the inclosing square. and angles at 4 and 6, up to forty-five degrees, may be laid off on the side 53. So angles at 3 and 1, up to forty-five degrees, may be laid off on the side 24; and angles at 2 and 4, up to forty-five degrees, may be laid off on the side 1 3. Angles for the front face may be laid off on all the sides of that face. But to lay off an angle at 6, on the line 3 4-although the measure 34 would make the angle 3 6 4 one of forty-five degrees, yet for lesser angles we should be obliged to take such proportions of 3 4 as the measures for angles, on the scale, bear to the entire side of the rectangular square from which the measures are taken. It would be the same when an angle at 4 or 3 should be required to be laid off on the side 1 2; or an angle at 1 or 2 should be required to be laid off on the side 3 4.

The same principles apply to the laying off of angles in semi-diagonal cabinet perspective. See pages 10 and 11 of Drawing-book No. IV. Yet, for practical purposes in all working drawings, the true angles, or inclinations of lines, can generally best be laid off by some known measurements on the objects themselves.

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* Note.—In the isometric ellipse, what is called the major axis (greater diameter) is a little more than once and seven tenths the length of the minor axis; and it is of the same length as the diameter of the circle which the ellipse represents. Thus, in Fig. 26, the upper ellipse represents a circle whose diameter is s t—that is, it represents the outer circle of Fig. 27, Plate VI., while the inner circle of Fig. 27 is the one we are obliged to compare it with in prescribing the rules of practical isometrical drawing. The reason of this is that the square within which the circle is drawn is diminished in apparent length of sides by an isometrical view of it; and we adapt the scale of our drawing to the apparent size, and not to the real size. Hence we draw a rectangular square, as A B C D, of Fig. 27, having the same real length of sides as the apparent length of the sides of the isometric square, A B C D, of Fig. 26; and then any lines, divisions, or points of the one may have corresponding lines, divisions, and points in the other. That is, both may be drawn to the same scale; and one may be used to illustrate the other. Thus the two kinds, cabinet and isometrical drawing, perfectly harmonize in measurement.

Taking, first, the upper face of the cube for illustration, we see that it is an isometrical square of ten spaces (ten feet) to a side, and crossed by equidistant isometrical lines. parallel to the sides. In Fig. 27, Plate VI., we have the rectangular square ABCD, of ten spaces (ten feet) to a side, and also crossed by the same number of equidistant lines parallel to the sides. A circle is also drawn within this rectangular square touching the middle points of its four sides, which circle is represented by the ellipse of Fig. 26. Now, as the inner circle of Fig. 27 is a circle of five spaces' radius, the circumference passes through the twelve numbered points of the intersections of the ruled lines, as there designated from 1 to 12 inclusive. (See page 150.) The ellipse of Fig. 26 must therefore pass through the same number of corresponding points in the ruling, so that we thus have twelve definite points through which the ellipse must be drawn. The ellipse may therefore, by these aids, be drawn quite accurately by the hand alone, by tracing a symmetrical curve through these twelve points. The same holds good as to the ellipses on the other two visible faces of the cube.

Any isometric ellipse that represents a circle of ten, twenty, thirty, forty, etc., spaces' diameter, may thus have twelve of its points given. But when the ellipses represent circles of other proportions, they must be drawn by the aid of the following principles and methods:

Scale of Diameters and Axes of Isometric Ellipses.

In every isometric ellipse there is, in addition to the major and the minor axis, what is called the isometric diameter. Thus, in the upper ellipse of Fig. 26, 1 7 or 4 10 is the isometric diameter of the ellipse-its position being centrally equidistant from, and parallel to, the sides of the inclosing isometric square. The isometric diameter is equal to a side of the isometric square. Hence, when an isometric square is laid down on isometrically ruled paper, the isometric diameter of the ellipse that may be drawn within it is also known, and may be located by merely counting the spaces on either of the side lines of the square.