Let m n be the lines, and a, a, the point. Through a, a, draw the

lines parallel to the given ones respectively. Let a B be the traces of the line parallel to n, n,, and Y, 8, those of the line parallel to m1 m. Then the traces of the plane through a, a, and which is parallel to m m and n1 n, are α, Y1, B2 8.

PROPOSITION XXI.

To construct the eidographic inclination of a given line to a given plane.

1. Through any point in the given line draw a plane perpendicular to the given plane (Prop. XVI.).

2. Find the intersection of this plane with the given plane (Prop.vII.). 3. Find the angle made by this line of intersection and the given line (Prop. XVIII.).

This is the angle required.

PROPOSITION XXII.

Through a given point to draw a line perpendicular to a given line.

1. Find the traces of the plane through the given point and given line (Prop. XIV.).

2. Find the traces of a plane perpendicular to the given line which shall also contain the given point (Prop. x., Scholium 2.).

3. The intersection of these two planes being formed, it is evidently the perpendicular required.

PROPOSITION XXIII.

To draw a line which shall be perpendicular to two given lines which are not in the same plane.

1. Through each of the given lines draw a plane parallel to the other (Prop. xx). These planes will be parallel.

2. Through each line draw a plane perpendicular to that which has been made to pass through the other line (Prop. xx11.).

3. Find the intersection of these perpendicular planes (Prop. VII.). This line will be perpendicular to both the given lines.

EXERCISES.

1. Construct the traces of a plane which passes through the three points (−1, 2, 3), (1, 3, 5), (2, 0, 3).

2. Given the horizontal projection of an equilateral triangle, and the vertical projection of one of its sides to construct the vertical projection of the entire triangle.

3. Given the horizontal and vertical projections of a tetrahedron to assign the conditions that must be fulfilled amongst the parts of the projection to render it a regular tetrahedron.

4. Through a given point to draw a plane parallel to two given lines. In what case does this problem become indeterminate?

5. Find a line equal to the distance between two given parallel planes.

6. Given the projections of a point and a line, to construct those of a line drawn through the given point perpendicular to the given line. 7. Through a given point to draw a line parallel to two given planes.

8. Draw a plane which shall make equal dihedral angles with two given planes:

(1.) The plane to be drawn through the intersection of the given ones;

(2.) Through any given point.

9. Given the projections of a finite line to divide it in any given ratio.

10. Find the angle contained between the traces of a given plane. 11. Draw a plane through a given line which shall make a given angle with either plane of projection; and another which shall make equal angles with both planes.

12. Given a trihedral angle and a point; to show the supplementary trihedral angle whose vertex is in the point.

13. Given the projections of any one radius of a circle to find the circle after revolution on the vertical plane.

14. Given the projections of three points in the circumference of a circle to construct its radius.

15. Through a given point to draw a line which shall meet two given lines.

16. Through a given point in a given plane to draw that line in the given plane which shall have the greatest inclination to the horizon, and that which shall have the least inclination to the vertical plane.

17. A building is to be erected on a declivity, but its foundation is to be horizontal, having the plan of the foundation, to trace on the surface of the ground the outline of the cutting.

18. A triangle on a plane is given: to find the position of a plane upon which if it be projected orthographically it shall become similar to a given triangle.

SECTION IV.

THE CONE, CYLINDER, AND SPHERE.

These are the only surfaces which are usually brought into engineering practice, except on very special occasions. When such occur, a more extended series of studies of the Descriptive Geometry must be entered upon, and the engineer must have recourse to treatises expressly and solely devoted to the subject.

The problems here given will amply display the general character of this class of researches; and it will be seen that they differ but little, as to general method, from those already developed at length.

It will be desirable, however, to prefix a few notes on the mode of designating surfaces to suit this method of construction.

(1.) As has been already shown, a plane is designated usually (because most conveniently) by its traces on two coordinate planes. Indeed one trace and a point on the other is sufficient; and when the plane is given by means of other specific conditions, it is still the best way of proceeding to find the traces of that plane as the preliminary step.

(2.) The cylinder being the extreme case of the cone (where the vertex is infinitely distant and the edges of the cone consequently parallel), the two surfaces will be designated by nearly similar methods, and the constructions will partake in a great degree of the same character.

(a) The cone and cylinder may each be defined (that is, their elements represented) by means

of the projections of two sec-
tions parallel to either of the
coordinate planes. Thus :-

cone.

When these two sections are
equal, the figure is a cylinder;
and when unequal, a
They are necessarily similar
whether equal or not (Prop. xx.
Chap. 1. Pls. and Sols.).

(b) The most usual way of

defining the cone and cylinder is by means of the horizontal trace of the surface and the projections of the axis. The horizontal trace (the vertical might have been taken instead) of the cone or cylinder is most frequently taken as a circle, in which case the centre of the circle is a point in the axis; and only one point more is required for fixing the axis. In the cone, that one point is invariably the centre of the surface or vertex; in the cylinder, any point whatever.

Even when the cone or cylinder is given by the first method, it is most convenient, and sometimes necessary, to reduce the conditions to the second method of definition.

(3.) The sphere is defined by means of its radius given in magnitude, and the projections of its centre given in position.

Sometimes it may happen that when the sphere, instead of being directly given, results from prescribed constructions, it may present itself by means of some other elements. Those elements can however always be, and indeed always are, reduced to the condition stated above, anterior to any further constructions which involve it.

Of the ellipsoid, which is the surface next to these as regards elementary character, nothing can be given in this course, and hence the mode of defining it may be properly omitted.

PROPOSITION I.

Given one trace and the projections of the axis of a cone to find the extreme limits of its projections on both planes.

Let v, v, be the vertex, e, the centre of the circular trace, and the circle as in the figure.

# कि

Draw the tangents a, a, b, b perpendicular to the axis; and join av, bv. The space av, b comprehends the projections on the vertical plane of all the points of the cone between the vertex and trace inclusive. If points beyond these limits be taken into consideration, they will still lie between the prolongations of these lines taken suitably to the case.

Draw the tangents v1 c1, v, d, to the trace. Then the horizontal projections of all the points in the cone between the vertex and the base inclusive will lie in the space bounded by the circular arc c, a, d, and the tangents c1 v1, d1 v. The extensions of the tangents c1 v1, di vi include the projections of all points in the extended cone, as in the former case.

In the cylinder the process is precisely similar, except that the limiting lines are parallel instead of converging to a point, as in the unlettered figure.

SCHOLIUM. The figures contain also the construction for the projections of the axis, which the student will explain.

PROPOSITION II.

Given the elements of a cone or cylinder and the projections of a point, to find whether that point be on the surface or not.

(1.) If the projections a1, a, of the point do not both lie within the

regions prescribed by the preceding problem, the point cannot be on the surface.

(2.) If they be situated within the limits, each will be the projection of some two points of the surface (since a line may cut the surface in two points); the horizontal projection being that of one pair of such points, and the vertical that of another pair of such points.

Suppose, then, that a, is the horizontal projection of a point on the surface: we are required to find whether a, is the vertical projection of the same point.

This

Draw v, a, (v, v, being the vertex) to meet the trace in P1, 91. is the horizontal projection of the edge of the cone through that point; and p1, q1 are the traces of the two edges, either of which might contain the point in question. Find, as usual, their vertical projections pv2, qu2, and let them cut a, a, in r and s. Then, if either r or s coincide with a,, a, will be the vertical projection of the point on the surface whose horizontal projection is a,: if not, the point a, a, is not upon the given surface.

SCHOLIUM. If a, fall between r and s, the point a, a, is within the cone or cylinder; if without, without.

In this process we have solved the problem: given_one_projection of a point situated on a given cone or cylinder to find the other.

There is however another case left for the student: when the vertical projection a, is given, to find a.

PROPOSITION III.

Through a given point to draw a tangent plane to a given cone or cylinder.

(1.) Let vv, be the given vertex of the cone, and a, a, the given point. Both these are in the tangent plane, and the tangent plane contains