Construct the plan of the square, and determine also the inclination of the plane on which it is situated.

Draw a b, a c, the two sides of the square, each 1.5 inch long, and forming at a an angle of 90°. Make the angle a b d equal to 34°, and the angle a ce equal to 20°. Draw a d at right angles to b d, and from a as a centre, with radius a d describe the arc dƒ, draw a ƒ through e at right angles to e c, and draw f g parallel to e c, and meeting a c produced in g. Draw the axis g b, produced towards k. This axis gb will be a horizontal or line of level on which the plane containing the figure is supposed to revolve. Complete the square by drawing c h and b h respectively at right angles to a c and a b, and construct the elevation by drawing x y at right angles to g bk. Make k m equal to a d, and draw k n parallel to a y. Draw the projectors a-1, c-2, and H-3 to x y, and parallel to g k. From centre m, with radius m-1, describe the arc 1-0, meeting k n in o, and from the same centre describe also the arcs 2-p and 3-q, meeting the line o q drawn through m, and determining likewise the real inclination of the plane containing the square, the lines drawn from o, p and q, parallel to g k, meeting lines drawn parallel to x y, in a a', c c', and h h', will determine the position of the corners of the square. (Fig. 39.)

Find the value of the angle x m o.

The following construction is a mere repetition of what has just been explained. The two adjacent sides a b and a c of a parallelogram are respectively 1 and 1.65 inches long, a b is inclined to the horizontal an angle of 38°, and a cat an angle of 16°. Find its projection and determine the inclination of the plane on which it is situated. (Fig. 40.)

A cube of 1 inch edge has two of its sides forming angles of 50° and 25° with the horizon, its base coincides with

that of a right square pyramid 3 inches high, showing on its surface at of its height lines parallel to its base-determine the projections of the cube and pyramid, showing also the intersection of the two solids.

It is evident that the first part of this problem is in every respect similar to those already described. On the line of inclination, o q, construct the elevation of the cube or, ps, m t, and q u, and determine likewise the height of the axis of the pyramid vw at of which at 7 draw a line at right angles to it, and determine the horizontal projection in the usual manner. (Fig. 41.)

Two sides 1.25 inches long of a triangular right pyramid 1.5 inches high, are respectively inclined at angles of 52° and 37° to the horizon, determine its projections. (Fig. 42.)

The preceding problems being understood, no difficulty whatever should be experienced in the construction of this and of the four following questions, which should require no further explanation, and be understood by mere inspection.

A triangular prism 1 inch side and .25 inch side, has two of the edges of its faces, forming respectively angles of 40° and 55° to the horizon, construct its projection. (Fig. 43.)

A pentagonal pyramid, 1 inch side, 2 inches high, has two of its edges inclined respectively at angles of 23 and 14° respectively to the horizon. Show its projections, and determine on its surface a horizontal, determined at its height taken on its axis. (Fig. 44.) A block 1 inch long, .75 inch broad, and .5 inch high, contains on its upper surface an ellipse whose major axis 1 inch long is inclined to the horizon at an angle of 50°, and its minor axis, .75 inch long, forms with the horizon an angle of 25°, construct its projections. (Fig. 45.)

A square block .8 inch side and .15 inch high, is surmounted by a cylinder .6 inch high, terminated by a cone

.65 inch high, whose base coincides with the top of the cylinder and whose surface is tangent to the sides of the block. Construct its projections. (Fig. 46.)

A plane inclined at an angle of 25° to the horizon has its horizontal forming an angle of 40° with the vertical plane. Construct its projections. (Fig. 47.)

On ay make the angle a b c equal to 40°: b c will be the horizontal required. Draw a c d at right angles to c b, and make the angle a c d equal to 25° (the angle which the plane makes with the horizon). Make a d at right angles to a c, and a e equal to a a at right angles to a b. Join e b.

APPLICATION.

The projections of a cube .85 inch edge standing on a plane inclined to the horizon at an angle of 25°, and whose horizontal trace makes an angle of 40° with the vertical plane. One edge of the cube making with this trace an angle of 50°: let the lowest face be considered likewise as the base of a right pyramid 1.65 inch high. (Fig. 48.)

The projections of the plane are determined in the manner that has just been explained.

Draw the side of the square CH at the angle of 50° with the horizontal trace b c, and project it on the eleva tion ca; reduce it now to its horizontal projection HC, and complete the plan of the square H C K L, and through the elevation on cd. Complete its projection in K' L', complete likewise on c d the profile elevation of the cube in m p, n q, o r, and c s, and construct its horizontal projection on G H' K' L' (the edges being at right angles to c b.) On c d determine the profile elevation of the square pyramid m n o c, having its apex in t, its horizontal projection will be determined by lines parallel to c b. To construct the elevation on the vertical plane, parallel to ca, draw the lines o o', n n', m m', and from a as a

centre describe the arcs m m'', n n'', o o'', and from m', n'', and o" draw lines parallel to a b. From every point in the figure horizontally projected, erect perpendiculars, cutting those lines at the intersection of which other lines drawn perpendicular to be will determine the position of the lines required to complete the elevation.

I have endeavoured thus far to reduce to certain fixed rules the method of Orthographic projections, and by constructing every given subject in the same manner, and according to the same principles, my aim was to demonstrate the analogy between each. There is nevertheless no fixed and universal method of projections, the results and principles are the same, but the methods vary. The diagrams 49 and 50 are examples to the purpose.

A cube .75 inch edge has one of its faces inclined to the horizon at an angle of 30°, construct its projections. Draw a b perpendicular to x y for an auxiliary plane, and construct the plane A, according to dimensions.

Construct the plane B at 30° for the inclination of the base of the solid. (Fig. 49.)

Horizontal and perpendicular projectors, as seen in the diagram, will first give the elevation B, and then enable the elevation C and horizontal projection D to be constructed.

A cylinder 1.25 inches diameter and 1.5 inches high, has its axis inclined to the horizon at an angle of 28°, construct its projections. (Fig. 50.)

Let the circle A B C D represent the plan of the cylinder. On AC as a base draw AE at the given angle (28°.) On AE as a base draw the elevation AG EF; the perpendicular FH, GK, will determine the minor axis of the upper elliptical surface, and LA that of the under face. The projection of the diameter BD, parallel to CK, will determine the breadth of the major axis: inspection will show the rest.

In mechanical and architectural drawing it is often necessary to represent the intersection of solids, or surfaces of different kinds, such as the junction of cylinders, pipes of different diameters, and at different angles, the crossing of arches, bridges, &c. These cases are solved in the following manner :

Two cylinders of equal diameters intersect in such a manner that their axes cross at right angles to each other. (Fig. 51.)

Inspection of the diagram will show that the intersection of cylinders of equal dimensions, and under these conditions will form in plan the cross a b c d.

The axes of two cylinders of different diameters, intersect at right angles; show their projection. (Fig. 52.)

Let a and b represent the vertical and horizontal projections of the cylinders.

Produce the points c e in c e' c e", and draw the semicircle GGG; and the points ff in f' f"; draw any where the horizontal d d parallel to c c, and cutting the circle in e e, and the semicircle in KH. Draw e e' e e'' parallel to ce' and c e'', take in the compass the distance HK, and transfer it from the axis C' C'' right and left to LL on the lines e' e''.

A curve drawn through fl c' l' will give the horizontal projection of one side of the intersection. The other side is to be completed in a similar manner.

The axis of two cylinders of different diameters intersect obliquely, show their projection. (Fig. 53.)

The inspection of the diagram will show that the principles of construction are precisely similar to the above.

APPLICATION.

The soffit, or intrados of an arched passage 32 feet wide, is intersected at right angles by an arched passage