the positions of the projections of any two points of that line are given on one plane of projection, and the positions of the projections of any two points of the same line, whether the same points as the former two or not, are given on the other plane of projection.

For the straight line will be in the plane passing through the first two points, perpendicular to the first plane of projection, and also in the plane passing through the second two points, perpendicular to the second plane of projection: its position will therefore be determined by the intersection of these two planes.

23. When the two straight lines ab, a'b' (fig. 11) are perpendicular to xy, and cut this line in different points m, n, they cannot be the projections of the same straight line. For through am let the plane amp pass, perpendicular to the plane xz, and through a'n the plane a'nq, perpendicular to the plane au. Then since the plane amp is perpendicular to the plane az, and that xy is perpendicular to am, the common intersection of the two planes, ay is perpendicular to the plane amp (Def. 6), and the plane amp is perpendicular to xy. In the same manner it appears that the plane a ng is perpendicular to xy. The two planes amp, a'ng being perpendicular to the same straight line xy are parallel to each other (Prop. 25); and consequently cannot contain any straight line common to both.

24. When the two straight lines ab, a'b' (fig. 12) perpendicular to xy meet this line in the same point m, then xy is perpendicular to the plane ama' (Prop.4), and therefore the plane ama' is perpendicular to each of the planes xz, xu (Prop. 32). Hence the directions of the projections of every line in the plane ama' will be in the lines ma, ma'; and if only these lines be given, the direction in the plane ama' of the line of which they may be the projections will be undetermined in order that it may in this case be determined, it is necessary that its projection a'n on some other plane tas, not passing through xy, should be given. But if a, a' the projections of one extremity of the line in the plane ama' be given, and likewise b, b' the projections of the other extremity, the line AB of which ab, a'b' are the projections will be given in magnitude and direction.

25. In general, the projections of any curve upon two planes which cut one another determine this curve.

Thus the curves abc and a'b'c' (fig. 13) being the projections of the curve ABC upon the planes xz, xu, conceive perpendiculars to the respective planes through the several points of the curves abc, a'b'c': they will form cylindric surfaces abcCAB, a'b'c'CAB, which must contain the curve ABC, and which consequently will determine this curve by their intersection.

If the cylindric surfaces do not cut each other, the given projections do not belong to the same curve in space.

26. A plane is determined when its traces upon two planes are given.

For two straight lines which cut one another determine the position of the plane passing through them (Prop. 2).

In general, the plane to be determined will cut the intersection ay of the two co-ordinate planes xz, xu; and it is evident that the point of meeting a (fig. 14) will be common to its two traces aa, d'a. If the plane be parallel to xy, its traces aa, a'a' (fig. 15) will also be parallel to xy (Prop. 23).

If the plane be perpendicular to xy, its traces will be perpendicular to xy (Def. 3).

If the plane be parallel to one of the co-ordinate planes, its trace upon the other plane will be parallel to xy (Prop. 27), and then this trace determines the plane.

If the plane pass through xy, its traces upon the two co-ordinate planes will be confounded in this line, which is not sufficient to determine the plane. In this case it is necessary to have its trace ba upon another co-ordinate plane sæt (fig. 16).

27. The preceding principles show how the data in problems in which the three dimensions of space are considered, may be represented accurately by points and lines situate in two fixed planes which cut each other, and which are inclined to each other at a convenient angle. In the figures, however, by which these principles have been illustrated, we have employed the ordinary mode of representation. To the imperfections of this mode of representation we have already adverted, and it remains to be shown how these principles are practically applied in the constructions of problems embracing the three dimensions of space. In this really consists descriptive geometry.

In the ordinary mode of representing geometrical magnitudes of three dimensions, and likewise in ordinary drawing, the eye viewing any object is supposed to be fixed, and the several points in the object are referred to an ideal plane between it and the eye, by lines drawn from the eye through these points, and intersecting this plane, which is that of the picture: the intersections of these lines with the plane of the picture, are the representations of the respective points of the object. This is the principle of Perspective Drawing.

Considering the orthographic projection of points as a mode of viewing them, the eye must no longer be considered to be fixed, but to be successively transferred over every point, so as to be in that perpendicular to the plane of projection which passes through the point, and we may consider the plane to which the points are referred to be beyond the object. If any series of points were thus orthographically viewed with reference to two planes, we should have an orthographic picture of them on each plane, and these two pictures properly combined, that is, in the two planes of projections, would, according to the foregoing principles, give these points in their true positions in space. It is evident that we may suppose

each of these pictures to be made separately on paper. If the line of intersection of the planes of projection be made common to the two pictures, so that one is immediately above the other on the paper, they will be combined in one drawing; and in order to have these orthographic pictures in their true position, it is only necessary that the paper should coincide with one of the planes of projection, that, for instance, which is represented below their intersection; and that the paper above this intersection should then revolve about this line until it coincides with the other plane of projection.

Thus, let ab (fig. 17) be the picture of a line thus orthographically represented on a plane here indicated by xyzw; and a'b' the picture of the same line orthographically represented on another plane indicated by xyuv; the two pictures having been separately made, but being here so combined that the line xy, representing the intersection of the planes of projection, is common to both. If the plane of the paper coincide with the original position of one of the co-ordinate planes xyz, the line ay coinciding with xy, the actual intersection of these planes, then if the paper above ay revolve about that line until the plane xyu coincides with the other of these planes, the projections ab, a'b' will be in their true positions, and the true position in space of the line of which they are the projections will be given by the intersection of planes passing through ab, a'b' and perpendicular to the respective planes of projection.

28. This manner of representing the two orthographic projections is in fact the same as the following. That all the constructions may be connected in a single drawing, one of the planes, xyu (fig. 18), for example, with the several lines and points on it, is made to turn about its intersection xy with the other plane xyz, until it coincides with the plane of xyz, in xyu. In this manner, the lines traced upon the plane xyz, not having changed, are in their proper position; but to have a correct idea of the true position of those in the other plane, we must imagine this plane replaced in its proper position. As to points out of these planes, they ought not to appear in the figures.

29. After the turning down of one of the planes of projection, the projections of the same point are connected in a manner which it is very important that we should notice. Let a, a' be these projections before the turning down of the plane xyu. Then, since the perpendiculars drawn from a, a on xy fall on the same point m (19); and during the turning of xyu about xy, the straight line a'm continues perpendicular to xy, when the plane ayu coincides with the plane of xyz in xyu', the line ma' becomes the prolongation ma" of ma. Hence this theorem, of almost continual use, that the projections of a point upon the two planes are in the same perpendicular to the intersection of the planes.

30. Although we have referred to the horizontal and vertical

planes as those usually employed in projections, we have hitherto made no hypothesis with respect to the inclination of the planes of projection. To render the constructions more simple, we shall now take these planes perpendicular to each other. The established custom is to consider one of the planes horizontal, although it may have any position whatever, and the other as vertical.

31. The intersection of the two planes is called the Line of Level. 32. The projections of any points, lines, or figures in space, on the horizontal plane, is called the Plan; and their projections on the vertical plane, the Elevation of these points, lines or figures.

33. A line or a plane is said to be horizontal or vertical according as it is parallel or perpendicular to the horizon.

34. Projections and traces are called horizontal or vertical according as they are in the horizontal or vertical plane.

35. Several consequences arise from the planes of projection being perpendicular to each other.

1. If a point or a line be in one of the two planes, its projection upon the other plane will be in the line of level.

For the perpendiculars which determine this projection are wholly in the first plane (Prop. 33).

2. If a line be in a plane parallel to one of the two planes, its projection upon the other plane will be a straight line parallel to the line of level.

For example, let a line be in a horizontal plane; then the perpendiculars which determine its vertical projection will be wholly in this horizontal plane (Prop. 33); and its vertical projection will be the intersection of this plane with the vertical plane, which is a straight line parallel to the line of level (Prop. 27).

3. If a plane be perpendicular to one of the planes of projection, its trace upon the other plane will be perpendicular to the line of level. For example, the vertical trace of a plane perpendicular to the horizon, that is, of a vertical plane, is perpendicular to the line of level (Prop. 34).

4. The perpendiculars drawn on the line of level, from the projections of a point, are equal to the distances of this point from the two planes of projection.

For a and a' (fig. 19) being the projections of the point A upon the planes xyz, xyu, when the planes are perpendicular, the angle ama is a right angle, and Aama' is a parallelogram. Therefore a'm Aa and am=Aa'.

When the vertical plane xyu is turned down upon the horizontal plane xyu', the line ama" drawn through the projections a, a" of the same point A is perpendicular to the line of level xy; and it here appears further that the part ma" gives immediately the elevation of this point above the horizontal plane, and that ma is its distance from the vertical plane.

36. When two planes intersect each other, they form, produced if necessary, four dihedral angles. Hitherto we have only considered the points projected to be posited in one of these angles; but it is evident that different points may be in any of the four angles, and it is important that we should examine how the representations of the projections of a point will be affected by the circumstance of its being posited in one or another of these angles.

Let qrwz, vtsu (fig. 20) be two planes perpendicular to each other, intersecting in xy, qrwz being horizontal and vtsu vertical, and forming the four dihedral angles uxyz, uxyq, sxyq, sxyz. Let the plane AA,AA be at right angles to the intersection xy of the planes, and therefore at right angles to each of the planes, intersecting them in aa, aa, Let aA, «A,, α,A,, aA be perpendicular to the plane vtsu, and equal to each other; and aA, aA a,A,, a,A perpendicular to the plane qrwz, and also equal to each other. Then A, A, A A will be corresponding positions of a point in each of the four dihedral angles; and the horizontal and vertical projections of A, A, A, A will be respectively a and a, a, and a, a, and a,, a and a,. When the vertical plane vtsu is turned round upon ay, the upper part xu down upon xq, and the lower part xs up upon xz, the horizontal and vertical projections of A, A, A A will be represented respectively by a and a, a, and a', a, and a', a and a' Representing these projections in their proper positions with respect to the intersection xy, a and a' (fig. 20. 1) are the horizontal and vertical projections of A; a, and a' (fig. 20. 2) the projections of A,; a, and a' (fig. 20. 3) the projections of A,; a and a', (fig. 20. 4) the projections of A To have them in their proper positions with respect to the horizontal and vertical planes, it is necessary that in fig. 20. 1, and fig. 20. 2, the plane xyu'v' should turn up on xy, as an axis, until it is perpendicular to the plane xyz; and that in fig.20.3, and in fig.20.4, the plane xys't' should turn down on xy, as an axis, until it is perpendicular to the plane xyq.

From this it follows, in the representation of these projections, that, taking any line ay on the paper to represent the line of level, the horizontal projections of all points which are in front of the vertical plane of projection being represented below the line of level xy, those of all points which are behind the vertical plane will be represented above that line; and the vertical projections of all points above the horizontal plane being represented above the line of level xy, those of all points below the horizontal plane will be represented below that line.

37. We now proceed to the solutions of the principal problems which may be proposed on the straight line and the plane.

In the figures which contain all the constructions of a problem, the data and the results will always be represented by full lines, and the lines in the construction by dotted or broken lines, which may be varied when necessary.