solid geometry, the student is advised to study closely the construction employed. Fig. 62a represents the HP and VP in position, with a line A B inclined to both, and its plan and elevation a b, a' b', the traces of the line being shown at HT and V T. If we fix our attention on A B, the line itself, and on a b, its plan, we can suppose that A B is held in position by the projectors A a and Bb, which pass from the ends of the line itself to the ends of its plan, perpendicular to the plan. Suppose,

further, that the line, its plan, and its projectors to the HP form a stiff frame, which can be turned about the plan of the line as a hinge, until it falls into the HP, as shown at A'a and B'b, then we have the true length of the line A B shown upon the HP at A' B', and we see at once that this true length is found by drawing perpendiculars from each end of the plan of the line, and making them equal in length to the distance of that end

above the HP. A similar reasoning applies to the line and its elevation a'b', together with its projectors to the V P, for if these be turned about the elevation a' b' as a hinge, until they fall into the V P, we shall have the true length of the line shown in the elevation exactly as in the plan. The figure also shows that if the true length of the line in plan be produced, it will meet the HT, and similarly in the elevation (Fig. 626), for by this construction we have produced the line to meet the two planes of projection, and we know the meeting points are the traces. The figure further shows that the real inclination of the line to the HP is the angle between the line itself, A B, and its plan ab, produced to meet at the HT, and that this is equal to the angle between the true length of the line A'B' and the plan ab-that is, the angle marked (theta);* and similarly the real inclination of the line to the V P is the angle between the line A B, and its elevation a' b' produced to meet at the V T, and that this is equal to the angle marked (phi), between the lines A" B" and a'b' (see Fig. 626).

*The construction for the true length of the line in the elevation is omitted in Fig. 62a for the sake of clearness.

PROBLEM XLIII. (Fig. 626).—Given the plan and elevation of a line to find its real length, its traces, and its inclination to the HP and VP.

a'

B"

V.T

6

Let a ba'b' be the given plan and elevation. From each end of the plan draw perpendicular lines equal in length to the height of the end above the HP, thus a A' equals height of end A, and b B' equals height of end B. Then A'B' is the true length of X the line, and its inclination to the HP is shown by producing A' B' and ab to meet in the H T, thus making the angle marked . Proceed in the same way from the elevation a'b', obtaining the true length A" B", the inclination

B

Α'

H.T.

Fig. 626.

to the VP shown by the angle and the V T. As a test of accuracy see that the real lengths in plan and elevation are the

same.

EXAMPLES.

EX. 7.-Draw the projections of a line A B, as in Ex. 6, and find its real length and its inclination to the HP and V P. EX. 8.-A is a point in the V P, 11′′ above the H P. B is a point in the HP, 13" from the V P. The real distance from A to B is 3". Draw the plan and elevation of the line joining A and B. (S. & A. Elem., 1887.)

EX. 9.-A point 1.5" from both planes of projection is distant 3.25" from another point, 2.25′′ from both planes of projection. Obtain the projections of the two points. (S. & A. E., 1888.)

EX. 10. Three equal lines 12" long, A O, B O, C O, meet at a point O at equal angles with each other. Draw the plan of the lines when neither of them is parallel to the V P, and consider them as the plan of three equal rods, 31" long, forming a tripod stand standing in the HP, then draw the elevation of the rods, and find their inclination to the HP.

PROBLEM XLIV. (Fig. 63).-To draw the projections of a line of given length inclined to both the HP and V P.

Let the line be A B, and its inclination to the HP and V P be and respectively.

From any point, A, in the X Y draw a line A b' equal in length to the given line A B, and making an angle with the

X Y equal to the inclination of the line to the HP. Then Ab' is the elevation and Ab the plan of the line, supposing it to be in the V P. Let the end b' remain in the V P and the end A be moved away from it, then so long as the end A moves in the HP its path must be in the semicircle, having b for centre and b A for radius, while the further it moves away from the VP the greater inclination will it have to the V P, and the shorter will its elevation become. If, then, we can determine what length its elevation will be, when its inclination to the V P is 9, we can draw its elevation, knowing that the position of the end b' has not altered. The last problem enables us to do this for we saw then that when a line is inclined to the V P and has one end in the plane, its elevation, its real length, and the

perpendicular from the end not in the plane, make a right angled triangle, of which the acute angle at the base is the angle of inclination to the V P. In the present case we know the hypotenuse of the triangle, the true length A b', and the acute base angle, the inclination, and we can, therefore, find the length of the elevation of the line. This is shown at b' A', the angle A 'A being made equal to and the angle at A' a right angle (see Prob. xliii.) Make, therefore, b'a' equal to b'A' and this will be the elevation of the line, its plan can be found by making the projector a' a equal to A A', for this we know is the distance of the end A. in front of the V P, or by dropping a projector from a' to meet the semicircle in a, then a b is the plan of the line.

a

Fig. 63.

EXAMPLES.

EX. 11.-Draw the plan and elevation of a line A B 32" long, inclined (a) 50° to the HP and 30° to the V P, (b) 25° to HP and 55° to V P.

EX. 12.-Draw the projections of a line C D 3" long, inclined 60° to the HP and 30° to the V P.

Projection of Plane Figures and Surfaces-Planes and

Traces of Planes.

Since plane figures and surfaces only possess two dimensions, length and breadth, it is only possible to draw their projections according to the principles of solid geometry, by supposing them to be contained by planes, the position of which relatively to the HP and VP of projection, coincides with the position of the figure. But in order that this principle may be appreciated, it is necessary to understand how planes are represented, since they are simply flat surfaces indefinite in extent, without thickness.

Planes. If the student will take a model of the planes of projection, such as a book open at right angles, and a set square to represent a plane, he will see that the plane can be placed in many different positions relative to the HP and V P. It can, for example, be placed so that its surface is perpendicular to both planes and touching both, or touching one and removed from the other; or the plane may be inclined to either the HP or V P, and have its surface at the same time perpendicular to the other, or the set square can be so placed as that its surface is inclined to both planes. It does not follow that the set square, or the supposed plane, will necessarily meet the HP and VP in the position in which it is placed; but since a plane is indefinite in extent, it is evident that if produced far enough it will somewhere intersect the HP and V P, unless parallel to them, and that the intersections will be lines, making certain angles with the ground line, depending upon the position of the plane. These lines of intersections are called the traces of the plane—that is, "the trace of a plane is its line of intersection with another plane.' The line where a plane intersects the HP plane of projection is termed its "horizontal trace," HT, and the line where it intersects the V P, its "vertical trace,” V T. Notice the distinction, that the trace of a line is a point, and the trace of a plane a line. From these considerations, as well as from the results of the little experiments first mentioned with the book and set square, we learn that a plane can only be shown by its traces.

Fig. 64 represents the traces of four planes in the only way in which they can be shown upon a flat sheet of paper, and with the help of a model the student should verify the following positions for the planes as shown:

Plane A.-Perpendicular to both the HP and V P.
Plane B.-Perpendicular to H P, inclined to V P.
Plane C.-Inclined to H P, perpendicular to V P.
Plane D.-Inclined to both HP and V P.

It will be seen that the traces are produced beyond the XY; this is the usual practice, and serves to show that the plane is not limited by the HP and V P.

With the help of Fig. 64, and of a model of the HP and V P, together with pieces of cardboard or set squares, to represent planes, the student should verify the following:

(a) A plane parallel to the HP or VP has no HT or VT respectively; therefore, when a plane is parallel to another plane, it has no trace upon that plane. Also, a plane parallel to one of the planes of projection is perpendicular to the other plane.

(b) When a plane is perpendicular to the HP its VT is a line perpendicular to the ground line; and when it is perpendicular to the V P, its HT is a line perpendicular to the ground line.

(c) When a plane is inclined to the HP, and perpendicular to the V P, its inclination is shown by the angle which its V T makes with the ground line; and similarly when a plane is inclined to the V P, and perpendicular to the H P, its inclination is shown by the angle which its H T makes with the ground line.

(d) When a plane is perpendicular to both the HP and V P, its traces form one straight line, perpendicular to the ground line.

(e) When a plane is inclined to both the HP and VP, its traces make angles with the ground line, which are not equal to the angles of inclination of the plane to the H P and V P.

(ƒ) The HT and V T of a plane intersect in the ground line, whenever the plane has two traces, except in the case where the plane is inclined to both the HP and V P, so that the sum of its inclination equals 90°, when its traces are parallel to the ground line.

(g) Parallel planes have parallel traces.

It should be noticed that a number of planes may be arranged relatively to the HP and VP in such a way as to intersect each other. In such cases the lines of intersection are represented by their plans and elevations.