described by the motion of a point, while surfaces are described by the motion of a line. 37. The surfaces of which we shall treat may be divided into-1, Surfaces of Revolution; 2, Conical Surfaces; 3, Cylindrical Surfaces. Surfaces of Revolution are those which may be supposed to be described by the revolution of a curve round its axis. Thus the semicircle revolving round its diameter produces the sphere, and the ellipse round its axes generates the spheroid. From this method of generation, it is plain that every section perpendicular to the axis is a circle; while every section made by a plane passing through the axis will reproduce the original curve. Conical surfaces are described by the motion of a line, which is compelled always to pass through a fixed point, and the other extremity moves through a curve traced upon a given plane. If the given curve be a circle, and if the given point be in a line passing through the centre of the circle, and perpendicular to its plane, the surface described is that of the common, or right, cone. Cylindrical surfaces are described by the motion of a line always parallel to its first direction, but the extremity moving through a given curve in a given plane. If the given curve be a circle, and the line be perpendicular to the plane of the circle, the surface described is the common, or right, cylinder. DEF. The moveable line which generates the surface is called the generatrix, and the line which directs and limits. its motion the directrix. THE TANGENT PLANE AND THE NORMAL. 38. We may consider a curve to be a polygon, of which the sides are infinitely small, and one of its tangents to be the prolongation of one of these indefinitely small sides. In the same manner a curve surface may be considered as a polyhedron, whose faces are infinite in number and infinitely small, and the tangent plane is the plane of one of these faces. And to have a clear idea of a tangent plane to a point in a surface, we must suppose that round this point there has been taken an infinitely small portion of the surface, which is considered a plane, and this plane, indefinitely extended, is the tangent plane. 39. If we conceive through the point of contact any number of lines whatever to be traced upon the surface, and if from the point under consideration we take infinitely small portions of these curve lines, we may regard these portions as straight lines, which produced will be tangents to the respective curves of which ultimately they form a part; but all these small lines will also be in the tangent plane, and therefore when produced will also be in that plane. Hence we see that the tangent plane to a surface contains all the tangents, drawn through the point of contact, to all the lines which can be traced through the point upon the surface. 40. This proposition is of great importance in determining the position of a tangent plane, since as it is only necessary to have two lines given in order to find a plane, it follows, that in order to construct the tangent plane to a given point of a surface, we have merely to draw tangents to two lines traced upon the surface, and to make the plane pass through these tangents. Our choice of the lines on the surface will, of course, be directed to those whose tangents are most readily found. DEF. A normal is a perpendicular to the tangent plane, drawn through the point of contact. TANGENT PLANES TO SURfaces. THE CYLINDER. PROBLEM 1. Given the horizontal trace of a cylinder and the direction of the generatrixes, to draw a tangent plane to a given point of the cylinder. 41. Let aecf be the horizontal trace of the cylinder; and in this instance suppose a ecf to be a circle. Draw ab, cd, tangents to aecf, and parallel to the horizontal projections of the generatrixes; these tangents will be also the limits of the horizontal projection of the cylinder, since all the generatrixes must be projected between these two lines. Draw the tangents ee', ff', to xy, and e'g', f'h', parallel to the vertical projection of the generatrixes; it is clear that these latter lines will be the limits of the vertical projection of the cylinder. Now, suppose that the horizontal projection m, of a point in the cylinder be given, and let the other projection be required. Draw through m a vertical plane, parallel to the generatrixes, which can intersect the cylinder only in one or more of m the generatrixes; the intersections of these generatrixes with the vertical, drawn through the point m, give the points of the surface which are projected in m. The traces of this plane are the line m q, drawn parallel to a b, through the point m, and the line qq' to xy. And since the first meets the curve aecf in i and k, it follows that the plane cuts the cylinder in two generatrixes, the horizontal projection of both being in the direction of mq; and their vertical projections are found by drawing i, kk', to xy, and i'q' k'q" parallel to f'h'. Draw mm' to xy; and m', m", where mm' meets i'q', and k'q', are the vertical projections of points of the cylinder corresponding to the horizontal projection m. This being done, next to draw a tangent plane to the point mm'. We see, first, that this plane ought to con tain the generatrix, i m, i'm', whose traces are i and q'; and farther, it must be a tangent to the cylinder at every point of this generatrix. Now the tangent plane to a surface contains the tangents to every curve traced upon this surface, and passing through the point of contact; if .. we draw the tangent ti, to the curve aec, it will be in the required plane, and since it is in the horizontal plane, it is the horizontal trace of the tangent plane. Produce it to meet xy in o; draw oq"; oq" will be the vertical trace of the plane. PROBLEM 2. To draw a tangent plane to a cylinder, through a point without it. 42. The tangent plane to the cylinder contains a generatrix, and in the preceding problem we have seen that its horizontal trace is a tangent to that of the cylinder: if, therefore, we suppose through the given point a line to be drawn parallel to the generatrixes, it will be altogether in the tangent plane; and if, after having constructed the horizontal and vertical traces of this parallel, we draw through the former, tangents to the horizontal trace of the cylinder, these tangents will be the horizontal traces of tangent planes, passing through the given point. Afterwards, to find their vertical traces, join the vertical trace of the same parallel to the points where the ground line is intersected by the horizontal traces of the planes. The details of the construction are represented in the next figure. The projections of the cylinder are traced, as in the preceding proposition; the projections of the given point are m and m'; those of the parallel to the generatrix are |