the nature of the surface, meet it in one of its right-line elements; but it will meet the plane of the base in a Fig. 31. straight line, as PT (fig. 31,) which will be a tangent to the periphery of the base.

If therefore we draw through the proposed point M, the straight line MP parallel to the generatrix AD, and construct the tangent PT to the point P of the curvilinear directrix, the tangent plane will be determined by the two lines PM and PT.

Remark 1st. If the proposed point through which the tangent plane is to be drawn, is without the cylinder, it is evident that there will be two planes which will answer to the conditions of the problem. To construct this solution find the point where a straight line passing through the given point and parallel to a rectilinear element of the cylinder, meets the plane of projection. A line drawn through this point, tangent to the intersection of the cylindrical surface with this plane, will be the trace of the tangent plane upon this plane of projection.

Remark 2d. If we would find the rectilinear tangent to a curve made by the intersection of two curve surfaces, we should construct tangent planes to these surfaces at the proposed point; and the intersection of the two planes would be the right-line tangent required.

86. PROBLEM. To construct a tangent plane to a

cone.

The only difference between this construction and that of the last problem, is, that the line MP must be drawn through the summit S instead of being parallel to Fig. 32. the generatrix (fig. 32).

Fig. 33.

87. PROBLEM. To draw a tangent plane to a surface of revolution.

The most convenient method of determining a tangent plane to surfaces of this class, is by means of the rectilinear tangents to the meridian curve and circular section, at the proposed point of contact.

The plane tangent to the point M (fig. 33) will therefore be determined by the straight lines MT and Mt; the first being tangent to the generating curve MX, and the other to the circle MZ. If therefore we can draw a tangent to the meridian curve at the proposed point of contact, it is easy to construct the plane tangent to the surface at that point.

The tangent plane to a surface of revolution, is always perpendicular to the meridian plane passing through the point of contact. It therefore follows that, a meridian plane to a surface of revolution is always normal to that surface.

Plane Sections of Curve Surfaces.

88. Plane sections of a cylindrical surface parallel to its axis and of a conical surface through its summit, are all straight lines (61, 63).

Plane sections of these surfaces parallel to the bases of these bodies are similar to these bases (El. 268, 275).

In the right cylinder and right cone these sections are perpendicular to the axis; if the bases are circles these sections will be circles. But an oblique section of a right cone by a plane which cuts all its rectilinear elements on the same side of the apex, is an ellipse (58). And it is easy to show to those acquainted with the properties of this curve, that a plane section of a right cylinder, oblique to its axis, is also an ellipse.

These sections are easily determined; for having these surfaces given, their rectilinear elements are known; and knowing these right lines and the cutting plane, the curves themselves are determined by finding the points where these lines meet this cutting plane, (33).

89. A plane section of a sphere is always a circle. The determination of these sections belongs to the solution of the general problem-To find the position and magnitude of the circle which is the intersection of a given sphere and a given plane.

By drawing a perpendicular from the centre of the sphere to the cutting plane, and determining the meeting of this line and the proposed plane, we have the centre of the circle required.

This solution will be very simple, if we take the plane of the vertical projections, DAB (fig. 34) perpendicular Fig. 34. to the horizontal trace of the proposed plane, which may always be done. Then, O' and Ó" being the projections of the centre of the sphere, if we suppose it cut by a vertical plane drawn through the line H' O' perpendicular to AC; this plane will pass through the centre of the

Fig. 34. sphere, and will contain the perpendicular drawn from this point to the proposed plane DAC; but it is parallel to the vertical plane DAB: We may therefore suppose it to be applied to this vertical plane without any change of the lines which it contains, either in magnitude or in position relative to its horizontal trace H'O' which will then coincide with AB. This being premised, M'E"N" will be the great circle which results from the section of the sphere by the vertical plane just mentioned; the perpendicular O"G" will determine the vertical projection G" of the centre of the section sought. We thence obtain its horizontal projection G'; and, revolving the plane DAC about it horizontal trace, as this centre will fall upon the point G in the horizontal plane, we may describe from this point, with the radius G"M", the circle MN which will be the required intersection of the sphere and plane proposed.

Of the Intersections of Curve Surfaces.

90. The most simple method of constructing the intersections of curve surfaces, is to suppose a series of plane sections of the two surfaces, and to determine the section made by each plane with each of these surfaces; the points common to the two curves, in any one of these plane sections, are of course points in the intersection required.

This method will be the most simple if we draw the cutting planes parallel to one of the co-ordinate planes. Suppose them parallel to the vertical plane; the sections which they make with the proposed surfaces will have their horizontal projections respectively in the horizontal traces of these cutting planes; and their vertical projections will not only show the height of any required point above the horizontal plane, but will be exactly equal to the auxiliary curves whose respective projections they

are.

A few examples will illustrate these principles, and enable the reader to apply them generally to the construction of analogous problems.

91. PROBLEM. To construct the intersection of a cylinder and a sphere.

We shall take for the plane of the horizontal projec

tions a plane perpendicular to the rectilinear elements of the cylinder; the vertical plane will therefore be parallel to these elements. We suppose cutting planes parallel to the vertical plane; their intersections with the cylindrical surface will be right-line elements of this surface; and their intersections with the sphere will be circles whose centres will have their vertical projection in the same point with that of the centre of the sphere. The radii of these circles may be easily found from the discussion in article 89.

The details of these constructions will be mostly left to the reader, as they will be found easy when the principles already discussed are well understood.

In the present case we draw straight lines G' I', g'ï', (fig. 35) in the horizontal plane, parallel to the ground line. Fig. 35. These lines are the horizontal traces of the auxiliary planes, which are supposed parallel to the vertical plane of projection; g'i is the radius of the circle which is the section made by one of these planes in the sphere whose centre has its horizontal projection in E' and its vertical projection in E". The points P' The points P', P', where the line g'i meets the base of the cylinder, are the horizontal projections of two rectilinear elements of this surface which are met each in two points by the circumference of the circle which is the section of the sphere by this auxiliary plane whose horizontal trace is g'i. If, therefore, from the point E" as a centre and with a radius equal to g'i, we describe a circumference, the points p" 1, P" 3, P",, P", where it meets the vertical projections of the rectilinear elements just mentioned, will belong to the intersection of the cylinder and sphere.

"

By a similar process we may determine as many other points as we wish of this intersection. There is, however, one circumstance in the general problem under discussion, which demands particular attention; this is the case in which the cylinder entirely penetrates the sphere, which it will do if the radius of a circular section of the cylinder is less than the radius of the sphere minus the distance of the axis of the cylinder from the centre of the sphere. In this case there will be two intersections of the surfaces in question, the one where the cylinder enters the sphere, and the other where it emerges from it. The horizontal projections of these two curves will of course be the same; and it will be readily seen that the

Fig. 35. parts P",, P"3, P4, P"2, belong to the vertical projection of the curve, where the cylinder enters the sphere; and

"

that p′′,, p′′з, P′′4, P"2, belong to the vertical projection of the curve where the cylinder emerges from the spherical surface.

The following figure represents the case in which the Fig. 36. distance of the axis of the cylinder from the centre of the sphere is greater than the difference of the radii of the sphere and cylinder. In this case the cylinder does rɔt entirely penetrate the sphere; there is therefore but one intersection of the two surfaces.

Fig. 37.

92. PROBLEM. To find the projection of the curve made by the intersection of the sphere and the cone.

Suppose the cone to be cut by planes passing through its apex, and perpendicular to the horizontal plane. The sections thus made in the conical surface, will be straight lines which are easily determined; the sections which these auxiliary planes make in the sphere, will be circles whose centres and radii are also easily found (89).

Let S'K' (fig. 37) be the horizontal trace of one of these cutting planes. Instead of revolving about this trace to coincide with the horizontal plane, we suppose it applied to the vertical plane DAB by bringing the line S'K' to AB, so that the point S' will fall upon S; then by taking Sk equal to S'K' we can draw the straight lines S ̋k, which will be the sections of the conical surface by the cutting plane. Making mg equal to S'G', the point g will be the position of the centre of the circle in which the sphere is met by the cutting plane, and whose radius is g h equal to G'H', (89).

The points p where the circumference of this circle mee's the straight lines S" k, belong to the intersection of the proposed cone and sphere. There are four of these points; two where the cone enters the sphere, and two where it emerges from it.

The points p are situated in the cutting plane. To have their projections, we must take S'P' equal to pi, and the point P' will be the horizontal projection; by drawing P'P" perpendicular to AB, the vertical projec tion P" will be found at the meeting of this straight line with pi; for the point p being taken in a vertical plane, is at the same height with its projection upon every other vertical plane.