revolved about the edge which separates the second face from the third, till this third face comes into the same plane. The process is continued, the body revolving upon each successive edge, as upon a hinge, till each of the faces has been in contact with the plane. The portion of the plane thus brought in contact with the prismatic or pyramidal surface, constitutes the developement of this surface. To apply this process to cylindrical and conical surfaces, we have only to consider the cylinder as a prism, and the cone as a pyramid, each of an infinite number of lateral faces. 69. If, instead of the right-line generatrix employed in the production of cylindrical and conical surfaces, we employ a curve line, the plane of which, in the progress of its motion, always cuts the plane of the curvilinear directrix in a line perpendicular to the tangent to the directrix at the point of intersection of the generatrix and directrix, the surface generated will be a surface of double curvature, 70. The most simple of double curve surfaces, are those which are also surfaces of revolution. Of surfaces of revolution we have, in The Elements, three examples: those of the right cone, the right cylinder, and the sphere. Of these the surfaces of the cone and the cylinder, are single-curve surfaces. The spherical surface is both a double curve surface and a surface of revolution. 71. If, instead of the semicircle used in the production of the sphere, we employ a semi-ellipse revolving about one of its axes, the surface thus generated will be a spheroidal surface; the body thus produced is usually called a spheroid, sometimes an ellipsoid. If the semiellipse revolve about its transverse axis, the product of its revolution is called a prolate spheroid; if the semiellipse revolve about its conjugate axis, it will generate an oblate spheroid. 72. If the curve-line generatrix be a hyperbola revolving about either axis of the curve as an axis of revolution, the surface generated by this motion will be a hyperboloidal surface; the body embraced by this surface begin a hyperboloid. 73. If the generatrix be a parabola revolving round the axis of the curve, the surface generated will be a paraboloidal surface, the body thus generated being a paraboloid. These three last bodies have been called conoids, on account of their being generated by the conic sections. These conoidal surfaces are surfaces of double curvature as well as surfaces of revolution. 74. There is manifestly an infinite variety of surfaces of revolution: But they all agree in one particular; namely, that every plane section perpendicular to the axis of rotation, gives a circular curve. A plane section of these surfaces through the axis, will always give the generating curve. This section is called the meridian section of the surface; or its meridian curve. 75. Another class of surfaces are called annular surfaces; these admit of great variety; the simplest is that of the common ring, which will serve to give a general idea of the class. This surface is generated by the motion of one circle along the circumference of a greater circle; the centre of the generating circle moving in the circumference of the directrix, with its plane at each point perpendicular to the tangent of this directing circumference. We In surfaces of this class, it is not necessary that either the generatrix or directrix should be circular, or that the directrix should be a curve of single curvature. Fig. 30. have, in figure 30, another example of this kind of surfaces; GH is the generating curve, and XZ is the curvilinear directrix. SECTION IV. Of the Projection of Curve Lines 76. It is evident that plane curves parallel to the plane of projection, will have, for their projection, curves equal to themselves. That is, the horizontal projection of a circle, parallel to the horizontal plane, is an equal circle: The horizontal projection of an ellipse parallel to the horizontal plane, is also an equal ellipse; and so of any other curve. But if plane curves are parallel to one of the co-ordinate planes they must be perpendicular to the other ; their projections, therefore, upon this other, will be straight lines. 77. To project, upon the horizontal plane, a plane curve parallel to it, perpendiculars are drawn to the plane from every point in the curve. These perpen diculars will form a cylindrical surface of which the proposed curve is a perpendicular section. And, in general, any curve line not perpendicular to the plane of projection, is projected by a cylindrical surface passing through the curve perpendicular to the plane of projection. This cylindrical surface is called the projecting cylinder of the curve. The projection is the intersection of this cylindrical surface with the plane of projection. 78. If the plane curve to be projected be inclined to the plane of projection, the projection will be a curve different from the proposed curve. To illustrate this principle, let us suppose a circle perpendicular to the vertical plane and inclined to the horizontal. One diameter of the circle will be perpendicular to the vertical plane and therefore parallel to the horizontal plane. The horizontal projection of this diameter will be a straight line equal to itself (22); and all the chords parallel to this diameter, will have their projected lines equal to themselves. But the diameter, perpendicular to that which is parallel to the plane of projection, will have the same inclination to this plane that the circle itself has to this plane; it will therefore have for its projection a line shorter than itself, whose magnitude depends upon the degree of inclination (22). So also the chords parallel to this diameter, will have for their projections straight lines bearing the same ratio to the chords, themselves as the projection of this diameter bears to the diameter itself. That is, the right-line elements of the circle taken in the direction of their greatest inclination to the plane of projection, have for their projections lines which bear a constant ratio to these ele ments. Now it is shown, in treatises on Conic Sections, that the straight lines, parallel to the conjugate axis, in an ellipse, have all the same ratio to the corresponding chords of the circumscribed circle, that is, the ratio of the conjugate to the transverse axis. The projection of a circle inclined to the plane of projection, is therefore an ellipse whose transverse axis is equal to the diameter of the circle, and whose conjugate axis is the projection of that diameter which has the greatest inclination to the plane of projection. This is sufficient to illustrate the principle, which the reader may apply to the ellipse and to other plane curves. 79. Curves of double curvature may also be determined by their projections upon two planes, as every point in the curve may be determined by its two projections. But those which result from the intersection of curve surfaces, will be determined by the projections of the intersecting surfaces. 80. Curves of double curvature have their parts frequently determined by the rectilinear tangents to points in these parts. The relation of a right-line tangent, to a curve of double curvature, is not easily explained. The rectilinear tangent to a plane curve, must not cut the curve and must be in the plane of the curve. Without this last condition an infinite number of straight lines might be drawn touching a plane curve in any point, conformable with the first condition: So also an infinite number of straight lines may be drawn through any point of a curve of double-curvature, without cutting the curve; yet only one of these can be a tangent to the curve. To determine the position of a rectilinear tangent to curves of this kind, the tangent has been defined to be "a straight line passing through two contiguous points in the curve." It is perhaps as philosophical to fix the position of the tangent by saying that, it must be in the plane determined by the point of contact and the two contiguous points of the curve. 81. A cylindrical surface is determined when any plane section of the surface and any right line element are determined by their projections; as the whole surface is composed of right-lines of each of which we have one point (namely, that in which it is cut by the plane making the given section) and the direction which is parallel to the given element. Any point in the surface may therefore be determined by article 23. A conical surface is determined by the projection of its summit and any curvilinear section, upon each of the co-ordinate planes. 82. A surface of revolution is most easily constructed by choosing one of the planes of projection perpendicular, and the other parallel, to the axis of revolution. Its projection upon the plane to which its axis is parallel, will be a meridian section. This, together with the projection of this axis upon the other plane, will enable us to determine every point in the surface. Of Tangent Planes to Curve Surfaces. 83. The theory of tangent planes to curve surfaces is important, as it enables us to determine the relative positions of these surfaces, when it could not be done by the surfaces themselves. A plane is tangent to a curve surface when it touches the surface, and lies wholly on the same side of the surface. Suppose a convex surface to be cut by planes all passing through the same point in that surface; the sections would be plane curves and the rectilinear tangents to these curves at their common point, would be all in the same plane. Any two of these right lines determine this plane; and this plane is tangent to the surface at this point. So that the construction of a tangent plane to any curve surface, reduces itself to the construction of two tangents to the curves in the surface at that point. 84. Suppose a straight line perpendicular to the tangent plane at the point of contact; this line is called a normal to the curve surface at that point: as also the normal to a plane curve at any point, is a line in the plane of the curve, drawn through the point of contact and perpendicular to the tangent. If through the normal to a curve surface, at any point planes be supposed to pass, cutting the surface, these planes will be normal to the surface at this point; they are called normal planes. 85. PROBLEM. To draw a tangent plane to a cylinder. A plane which touches a cylindrical surface, must, from |