THEOREM XVI.

If two straight lines be parallel to one another, their projections on the same plane shall also be parallel.

Let AB and CD be parallel; it is required to prove that their projections ab, cd on MN are parallel.

Proof. Because Aa and Cc are both perpendicular to the same plane MN (Def. 10), they are parallel to one another, (Theor. VII.).

......

Therefore, the plane aA Bis parallel to cCD,... (Theor.XIV.). But ab, cd are the common sections of these two parallel planes with MN.

un

If two straight lines be at right angles to one other, their projections on a plane parallel to any one of them shall also be at right angles.

Let the lines AB, CD be at right angles to one another, and the plane MN parallel to AB; it is required to prove that the projections ab, cd are also at right angles.

Proof. Because AE is parallel to MN it is parallel (Theor. X1).

to de,

E

And since Ee is at right angles to ae, it is also at right angles to AE.

Therefore AE is perpendicular to the plane CDd, since it is perpendicular to the two lines CD, Ee, (Theor. III.).

......

But ae being parallel to AE is also perpendicular to the plane CDd, and consequently to the line cd. That is the angle aed between the projections of AB, CD is a right angle,

THEOREM XVIII.

If the projections of two straight lines be at right angles to one another, and one of the lines parallel to the plane of projection, the two lines shall be at right angles to one another.

If ab and cd be at right angles and AB be parallel to the plane MN, then AB and CD are at right angles.

Proof. Since ae is at right angles to Ee and ed, it is a normal to the plane CDd.

But AE is parallel to MN, and therefore also to ae, ...... (Theor. XI.).

Hence AE is normal to CDd, ...... (Theor. VIII.).

Therefore AB is at right angles to CD, ...... (Def. 5).

CHAPTER II.

INTRODUCTION TO DESCRIPTIVE GEOMETRY AND PROBLEMS ON THE STRAIGHT LINE AND PLANE.

DESCRIPTIVE GEOMETRY has for its principal objects the representation of solid figures on a plane surface, and the graphic solution of the problems of Solid Geometry.

In other words, it is that branch of Geometry by means of which accurate drawings of machines and structures are made, and problems respecting solid figures reduced to those of Plane Geometry.

As an illustration of the use and importance of this branch of Geometry, suppose the position of a plane and point in space to be known, and that it is required to find the distance of the point from the plane. This will require the drawing of a perpendicular from the point to the plane, finding the point of intersection, and then determining the distance between the two points. Euclid shows how a perpendicular from a point to a plane may be drawn, but in solving the problem practically according to his directions it would be found necessary to work on three different planes, the relative positions of which cannot be determined beforehand, but must be fixed at the different steps of the construction. In attempting to solve this elementary problem by Euclid's method, the necessity will soon be felt of adopting some mode of construction in which the data of the problem may be represented and the necessary con

structions executed on a single plane surface. Both these advantages are obtained by the method of projections (see defs. 10 to 12), which is the method used in Descriptive Geometry.

A point is completely determined when its projections are given on two intersecting planes, the positions of which are known; for there will then be two lines-the projectorson which the point must lie, and it will consequently be their point of intersection. Let a and a' be the projections of a point on the two planes H and V (fig. 1), then the point must be in each of the lines Aa and Aa'; so that there is only one point A which can have the projections a and a'.

A line, straight or curved, is completely determined when its projections are given on two intersecting planes; as will be seen from fig. 3, where in one case the line AB is the common section of the two projecting planes ABab, ABa'b', and in the other the line BCDE is the common section of the two projecting surfaces (def. 12) BCDEedcb, BCDEe'd'c'b'. The only exception would be when the two projecting planes of the line coincided, that is when they were perpendicular to the common section of the two fixed planes (Theorem VI.), and in that case it would be necessary to have a projection of the line on another plane not parallel to either of the first two.

These fixed planes to which points and lines are referred are called co-ordinate planes or planes of projection. They are taken at right angles to one another and one of them is supposed to be horizontal and the other vertical; thus (H) in fig. 1 is the horizontal plane of projection, and (V) the vertical plane of projection. Their common section xy is called the ground line.

Projections on the horizontal plane are called horizontal projections, and those on the vertical plane vertical projections. The equivalent terms plan and elevation have been long in use in connection with the drawings of buildings and other objects, and are now frequently extended to the projections of points and lines.

As the planes are at right angles to one another any point on one of them has for its projection on the other a

point on the ground line (Theor. v.); thus in fig. 9, a' and b, on the ground line, are the projections of the points A and B respectively, one in each plane of projection.

The plane which contains the two projectors of a point, as a Aa fig. 1, must be perpendicular to each plane of projection (Theor. IV.), and consequently to the ground line (Theor. VI.), so that xy is at right angles to aa。 and a'a: hence

The perpendiculars drawn from the projections of a point to the ground line meet it in the same point.

The lines aa, and a'a, are respectively equal to the projectors Aa and Aa; that is aa, and a'a, are equal to the distances of A from the planes of projection.

d

Fig. L

a'

(H)

Fig. 2.