If the surface of a solid is neither horizontal or perpendicular it is
oblique.
Thus, if we place a box on the table, the top of the table represents
the horizontal plane and the side of the box the vertical plane, and
the intersection of the box and the table is the ground line, or X Y.
Draw a line out square from the box on the table, marked N N, Fig. 1.
Hold the end of a book on this line, with its edge against the side of
the box in an inclined position; mark a line on the side of the box:
this line is the vertical trace, because the oblique plane has cut the
vertical plane on this line. Before moving the book mark a line on
the table: this is the horizontal trace, for the same reason that the
oblique plane has cut the horizontal plane on this line.
Fig. 2 shows the horizontal trace inclined 60° to the vertical plane.
Draw the dotted line N N on the table, making an angle of 60° with the
box. Place the end of a book on this line, and while in an inclined
position mark the vertical and horizontal traces the same as in Fig. 1.
To find the true inclination of the oblique plane take F for centre,
and for radius F R, strike the arc to cut X Y in P; join E P, which
is the true length and inclination of a line on the oblique plane, to
stand vertically over F R. All lines on the oblique plane parallel to
the horizontal trace will be level, and all lines square to it will be
the true inclination of the surface of the oblique plane.
Fig. 3. It is required to cut a block of wood the size of the square
A B C O, its side A B to be 5 inches high, and its top surface inclined
30° to the horizontal plane. On the oblique surface project an ellipse
that will stand vertically over the quarter of circle on plan. Let
A B C O be the plan and B C R N the elevation. Project 7, 8, 9 on to
the oblique surface, as shown by 1, 2, 3.
Fig. 4 shows a cuneiform sketch of the block. Make B N and C R equal
corresponding letters, Fig. 3; join R N; square out lines from N R;
make N A´, R O´ equal A B, Fig. 3. To complete the figure, make A A´
equal B N, and O O´ equal C R. To draw the ellipse, make N 1 2 3 R
equal N 1 2 3 R, Fig. 3. Make 1 7 and 2 8 and 3 9 equal 4 7 and 5 8 and
6 9, Fig. 3. Trace the curve through A´ 7 8 9 R as shown. If this block
is cut out in a vertical direction to the ellipse on its surface it
will stand correctly over the quarter of circle, its plan.
Fig. 5. Cut a block of wood so that its edge will stand vertically over
A B C O. The top of the block to be hard down at A. From A to B rise
3 inches, and from B to C 4 inches more. Make B F equal 3 inches and
C 5, 7 inches. Join 5, F extended to cut X Y in E. Join E A, which is
the horizontal trace, and E F 5, the vertical trace. F 5 will be the
inclination of the edge of the block over B C. To get the length and
inclination of the edge over A B, take B for centre and B A for radius,
strike an arc to cut X Y in H. Join F H for the required edge.
The process of getting the lines on the oblique surface of this block,
as shown at Fig. 6, is the same as most of the face moulds as laid down
in this book, and let it be understood that if the problems on this
and the following Plate are properly mastered, the foundation upon
which this system depends has been laid, and all the plates that follow
are purely a matter of detail. Let the instructions given here be
carried out, and cut the blocks of wood as described, when the meaning
and intention of every line will be clearly illustrated, and the way
cleared for further progress.
Make E F 5 equal E F 5, Fig. 5. Take the distance F H, Fig. 5, in
the compasses, with F, Fig. 6, as centre, strike an arc at A. With
E A, Fig. 5, as a radius, and E, Fig. 6, as centre, strike an arc to
intersect the first one at A. Join E A for horizontal trace and F A
for the top edge of the block over A B. Draw from 5 parallel to F A,
and from A parallel to F 5. Then O will be the centre of the ellipse,
as it will be vertical over the centre on plan. A line drawn on an
oblique plane square to the horizontal trace and passing through its
centre is the major axis, and a line drawn parallel to the H T and
passing through its centre is the minor axis. Make 2 3 0 5 equal
2 3 0 5, Fig. 5. Make 3 6 and 0 7 equal 8 6 and 0 7, Fig. 5, and trace
the curve through A 6 7 5.
[Illustration: PLATE 13.]