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Next, as A is elevated above Cc a distance equal to AH or AF, the point A' must revolve until it is this distance above xy, for the plane revolves upon the horizontal Cc, which is represented in elevation by C'. Therefore make c'd' equal to AH or AF, and draw e'd parallel to æy, cutting the arc in é'. Join é'C' and produce it to f', then ef is the elevation of the plane containing the triangle.

Further, project B to B, and with centre C', and radius C'B', describe an arc, cutting ef g'. Through A and B draw unlimited lines parallel to xy, and from points é and g' draw em, g'n, cutting

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these lines in m and n. Join mn, nC, Cm, then mnC is the plan of the triangle when the sides AB and ÁC are inclined at 60° and 30° respectively.

Problem 32. The traces of a plane being given, to find the angles which it makes with the planes of projection.

Let AB and AC be the traces of the given plane. It is required to find the angles which it makes with the planes of projection.

First, draw any line ED at right angles to the horizontal trace AB, and draw DD' perpendicular to xy, meeting the vertical trace AC' in D. Now, since AC intersects the vertical plane, point D' will be the vertical trace of a line in space whose horizontal projection is ED.

Secondly, the angle which ED makes with the horizontal plane may easily be found. This agle is DOD or GED. Now, ED is the projection of EG, and since the line and its projection are drawn

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from the same point E, which is common to the horizontal plane and the given plane, it follows that GED is the angle which the plane makes with the horizontal plane of projection.

Thirdly, the angle which the given plane makes with the vertical plane of projection is FK'D, and it is found by drawing K'D perpendicular to AC', and constructing the right-angled triangle FR'D, as has been previously done, viz., by drawing DF at right angles to DK', and making it equal to DH.

NOTE.—When the traces of a line are situated in the traces of a plane, the line is said to lie in the plane. Thus, the line joining E and D lies in the plane whose traces are AB and AC'; and when one projection, as ED, is given, the other projection may be found.

Problem 33. To determine by its traces the plane containing three given points.

Let ABC and ABC be the projections of the given points. It is required to determine a plane which shall contain them.

Now, if two points are contained by a plane, it is evident that the line joining those two points must also be contained by that plane. Further, if a line be contained by a plane, the traces of that line are in the traces of the plane. The knowledge of these two principles is sufficient for the solution of this problem, for the required plane must contain each of the three lines AB, BC and AC, the traces of which will be points in the traces of the plane,

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Join A'B', AB, B'C, BC, and produce B'A' beyond A' to meet xy in D. Then a perpendicular to .cy through D', intersecting the plan of AB produced in D, gives one point in the required horizontal trace, and E, which is the horizontal trace of the line BC, is a second point in that trace. The line DF drawn through these points is the horizontal trace of the required plane. Find Go, the vertical trace of the line AB, and draw H'F through Gʻ to meet DF in F. Then HFD is the plane containing the three given points.

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Section V.

FURTHER PROJECTIONS OF SOLIDS.

In addition to the various solid figures already referred to in the previous sections, there are four other regular solids which must be mentioned, viz.

(A) The Tetrahedron, which is a solid figure contained by

four equal and equilateral triangles” (Euc. XI., Def, 26).

(B) The Octahedron, which is “ a solid figure contained by eight

equal and equilateral triangles" (Euc. XI., Def. 27).

[graphic]

(C) The Dodecahedron, which is “a solid figure contained by twelve equal pentagons, which are equilateral and equiangular" (Euc. XI., Def. 28).

[graphic]

(D) The Icosahedron, which is “a solid figure contained by

twenty equal and equilateral triangles” (Euc. XI., Def. 29).

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To construct the projections of a cube having a face and one of its edges inclined at given angles.

Let the face ABCD be inclined at an angle 0 to the horizon, and the edge BC at an angle '; being greater than 0.

On xy, a line of level perpendicular to the trace min of the plane of the base, make an elevation np of this plane ; in it place the line Bm, whose plan is bm inclined at an angle 8; construct the elevation

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