FIG. 524.—A rectangular sheet lying horizontally and carrying four vertical pins. No edge of the sheet is parallel to the Picture Plane.

E.L.

C.V.

b

FIG. 525. A rectangular sheet whose plane is vertical, but not parallel to Picture Plane. Two edges are parallel to Picture Plane.

E.L. C.V.

e

d

FIG. 526.-Two intersecting rectangular sheets. The edges ac, fd, gl, hk are all | Picture Plane, and the remaining edges

to it.

(3) The length of the picture of a straight line depends not only on the length of the line but also on (i) the distance of the line from the eye, and (ii) the direction of the line relative to the line of sight.

Since abcd (Fig. 523) is the picture of a rectangle, ab and dc are pictures of equal lines, but ab is shorter than de because the corresponding line is farther from the eye. Also be and ad are pictures of equal lines, but be is shorter than ad because the corresponding line, though nearer to the eye, lies more nearly in the line of sight. If a line points directly towards the eye, its picture obviously shortens to a point.

(4) If two horizontal lines are parallel their pictures converge to a point in the eye-level (unless the lines are parallel to the Picture Plane). For example, ab and dc in Fig. 524, ba and cd in Fig. 525, also da, cb, he, gf in Fig. 523. In Fig. 523 ab and dc do not converge because they represent lines parallel to the Picture Plane.

(5) If a line is perpendicular to the Picture Plane its picture points toward the Centre of Vision. See the lines da, cb, he, gf in Fig. 523, and the lines fa, eb, dc, hg, kl in Fig. 526.

Caution. The picture of an angle is usually an angle, but these two angles are usually unequal. The picture of an angle is a line, if the eye lies in the plane of the angle.

EXERCISES CXXXV.

1. Hold two pen-holders at right angles and, considering a sheet of paper as a Picture Plane, hold these pen-holders in front of the sheet so that the angle between them in their picture will be (a) a right angle, (b) zero, (c) 180°, (d) about 45°.

2. Take a piece of note paper with the two leaves opened out at an angle of about 45°. Stand it in various positions and sketch it.

3. Take an empty match-box and sketch the two portions separately in various positions.

4. A straight piece of rail-road contains two pairs of lines laid on sleepers in the usual way, and runs due north. Sketch roughly the appearance of the rails and the outlines of the sleepers (i) to a man standing on a platform and facing due east, (ii) to a man standing between the two pairs of lines and facing due north.

5. Prove that a straight line cannot meet a plane at more than one point, unless it lies altogether in the plane.

6. Of three straight lines, not in the same plane, each meets the other two. How is this possible?

7. Into how many regions is space divided (a) by two parallel planes and a third plane intersecting both? (b) by two pairs of parallel planes? 8. Three faces of a rectangular block meet at one corner; into how many regions is space divided by the three complete planes containing these three faces?

395. Intersections.-Two straight lines cannot intersect at more than one point. If they do not intersect they are not necessarily parallel, as they may not lie in the same plane.

For example if a line is drawn on the table pointing north and another line on the floor pointing north-east these two lines do not meet and are not parallel. If, however, both lines point north (or both north-east) they are parallel and a plane can be imagined which contains both.

A straight line and a plane cannot intersect at more than one point. If they do not intersect they are parallel (by Definition).

Two planes can only intersect in a straight line (see § 396). If they do not intersect they are parallel (by Definition).

396. THEOREM.-If two planes meet their intersection is a straight line.

Let XX, YY be the two planes, and let A and B be any two points on their line of intersection (Fig. 527).

X

FIG. 527.

lie in plane XX,

.. the straight line AB lies in

plane XX.

Def.

Similarly the straight line AB lies in plane YY.

Again the two planes cannot meet in any point C which is

not in AB, for only one plane can pass through the three points A, B, C.

§ 391.

Hence the intersection of the two planes is the straight line AB.

COROLLARY.-If a point lies in each of two planes it must lie on their line of intersection.

397. THEOREM.-If three planes intersect in pairs their three lines of intersection are coincident, concurrent, or parallel.

Let P, Q, R be the three planes, each of which meets the other two.

Then there are three lines of intersection, viz. where Q meets R, where R meets P, and where P meets Q.

CASE I. When P intersects Q and R in the same line XY (Fig. 528).

Then since the line XY lies in each of the planes Q and R, it is the intersection of Q and R.

Thus the three lines of intersection are coincident.

CASE II.-When P intersects Q and R in two intersecting lines HK and LM respectively (Fig. 529).

Let Q and R intersect each other in the line FG. Let HK and LM intersect at a point / (not shown in the figure).

./ lies in HK and HK lies in plane Q,

../ lies in plane Q.

:.:/ lies in LM and LM lies in plane R, ../ lies in plane R.

/ lies in both the planes Q and R,

.. it lies in their line of intersection FG.

§ 396.

Thus the three lines FG, HK, LM are concurrent, meeting at the one point .

CASE III. When P intersects Q and R in two parallel lines HK and LM respectively (Fig. 530).

Then FG is parallel to both HK and LM.

If not, if possible let FG meet LM at a point (not shown in the figure).

Then, arguing as in Case II, it can be proved that / lies in each of the planes P and Q, and therefore lies in their intersection HK.

Thus HK and LM meet at ; which is impossible since they are parallel.

Hence FG is parallel to LM.

Similarly it can be proved that FG is parallel to HK. Hence the three lines of intersection are all parallel.

EXERCISES CXXXVI.

1. Into how many regions is space divided by two intersecting planes?

2. Into how many regions is space divided by three planes which meet in one straight line?

3. Into how many regions is space divided by three planes which meet in three parallel straight lines?

4. Into how many regions is space divided by three planes which meet in three concurrent straight lines? [Work this by starting as in Ques. 1.]