the line CD at the points C, F, and D, by the parallel planes M, N, and P, then

579. The general problem of perspective in drawing, consists in representing upon a plane surface the apparent form of objects in sight. This plane, the plane of the picture, is supposed to be between the eye and the objects to be drawn. Then each object is to be represented upon the plane, at the point where it would be pierced by the visual ray from the object to the eye.

All the visual rays from one straight object, such as the top of a wall, or one corner of a house, lie in one plane (60). Their intersection with the plane of the picture must be a straight line (516). Therefore, all straight objects, whatever their position, must be drawn as straight lines.

Two parallel straight objects, if they are also parallel to the plane of the picture, will remain parallel in the perspective. For the lines drawn must be parallel to the objects (572), and therefore to each other.

Two parallel lines, which are not parallel to the plane of the picture, will meet in the perspective. They will meet, if produced,

at that point where the plane of the picture is pierced by a line from the eye parallel to the given lines.

EXERCISES.

580.-1. A straight line makes equal angles with two parallel planes.

2. Two parallel lines make the same angle of inclination with a given plane.

3. The projections of two parallel lines on a plane are parallel. 4. When two planes are each perpendicular to a third, and their intersections with the third plane are parallel lines, then the two planes are parallel to each other.

5. If two straight lines be not in the same plane, one straight line, and only one, may be perpendicular to both of them.

6. Demonstrate the last sentence of Article 579.

TRIEDRALS.

581. When three planes cut each other, three cases

3d. The three intersections may meet at one point.

through the intersection of the first two, it will divide each of the diedrals into two parts, making eight in all. Each of these parts is called a triedral, because it has three faces.

A fourth case is impossible. For, since any two of the intersections lie in one plane, they must either be parallel, or they meet. If two of the intersections meet, the point of meeting must be common to the three planes, and must therefore be common to all the intersections. Hence, the three intersections either have more than one point common, only one point common, or no point common. But these are the three cases just considered.

582. A TRIEDRAL is the figure formed by three planes meeting at one point. The point where the planes and intersections all meet, is called the vertex of the triedral. The intersections are its edges, and the planes are its faces.

The corners of a room, or of a chest, are illustrations of triedrals with rectangular faces. The point of a triangular file, or of a small-sword, has the form of a triedral with acute faces.

The triedral has many things analogous to the plane triangle. It has been called a solid triangle; and more frequently, but with less propriety, a solid angle. The three faces, combined two and two, make three diedrals, and the three intersections, combined two and two, make three plane angles. These six are the six elements or principal parts of a triedral.

Each face is the plane of one of the plane angles, and two faces are said to be equal when these angles are equal.

Two triedrals are said to be equal when their several planes may coincide, without regard to the extent of the planes. Since each plane is determined by two lines, it is evident that two triedrals are equal when their several edges respectively coincide.

583. A triedral which has one rectangular diedral, that is, whose measure is a right angle, is called a rectangular triedral. If it has two, it is birectangular; if it has three, it is trirectangular.

A triedral which has two of its faces equal, is called isosceles; if all three are equal, it is equilateral.

SYMMETRICAL TRIEDRALS.

584. If the edges of a triedral be produced beyond

the vertex, they form the edges

of a new triedral. The faces of these two triedrals are respectively equal, for the angles are vertical.

Thus, the angles ASC and ESD are equal; also, the angles BSC and FSE are equal, and the angles ASB and DSF.

E

S

B

The diedrals whose edges are FS and BS are also

equal, since, being formed by the same planes, EFSBC and DFSBA, they are vertically opposite diedrals (555). The same is true of the diedrals whose edges are DS and SA, and of the diedrals whose edges are ES and SC.

In the diagram, suppose ASB to be the plane of the paper, C being above and E below that plane.

But the two triedrals are not equal, for they can not be made to coincide, although composed of parts which are respectively equal. This will be more evident if the student will imagine himself within the first triedral, his head toward the vertex, and his back to the plane ASB. Then the plane ASC will be on the right hand, and BSC on the left. Then let him imagine himself in head toward the vertex, and his back to the plane FSD, which is equal to ASB. Then the plane on the right will be FSE, which is equal to BSC, the one that had been on the left; and the plane now on the left will be DSE, equal to the one that had been on the right.

the other triedral, his

Now, since the equal parts are not similarly situated, the two figures can not coincide.

Then the difference between these two triedrals consists in the opposite order in which the parts are arranged. This may be illustrated by two gloves, which we may suppose to be composed of exactly equal parts. But they are arranged in reverse order. The right hand glove will not fit the left hand. The two hands themselves are examples of the same kind.

585. When two magnitudes are composed of parts respectively equal, but arranged in reverse order, they are said to be symmetrical magnitudes.

The word symmetrical, as here used, has essentially the same meaning as that given in Plane Geometry (158). Two symmetrical plane figures, or parts of a figure, are