It will he well to place before the reader here a clear statement of the class of problems he may expect to meet with under this head. It will include only the envelopes of such solid figures as have for a base the circle, or any figure of equal or unequal sides which may be inscribed within a circle, and which terminate in an apex located directly over the center of the base.

According to the definition of an inscribed polygon (Def. 66), its angles must all lie in the circumference of the same circle. So the angles or hips of a pyramid whose base can be inscribed in a circle must lie in the surface of a cone whose base circumscribes its base and whose altitude is equal to that of the pyramid. Therefore the circle which describes the pattern of the base of the envelope of such a cone will also circumscribe the pattern of the base of the pyramid contained within it. The envelopes of such solids. therefore, as scalene cones, scalene pyramids and pyramids whose bases cannot be inscribed within a circle are not adapted to treatment by the methods employed in this section. Even the envelope of an elliptical cone cannot be included with this class of problems because it possesses no circular section upon which its circumference at any fixed distance from the apex can be measured.

In this connection it is proper to call attention to the difference between a scalene cone and a right cone whose base is oblique to its axis. According to Definition 96, a scalene cone is one whose axis is inclined to the plane of its base, and according to Definition 94 the base of a cone is a circle. As any section, of a cone taken parallel to its base is the same shape as its base, any section of a scalene cone taken parallel to its base must be a circle, and any section taken at right angles to its axis could not, therefore, be a circle, but would be elliptical. Again, as any section of a right cone (Def. 95) at right angles to its axis is a circle, if its base be cut off obliquely, such base would. according to Definition 113, be an ellipse. Therefore, since its horizontal section is a circle, its envelope may be obtained by methods employed in this section. (See Problem 136.) And since the section of a scalene cone taken at right angles to its axis is an ellipse, the scalene cone becomes virtually an elliptical cone with an oblique base - that is, with a base cutoff at such an angle as to produce a circle - and, as stated above, cannot be included in this section.

The principles governing the problems of this section are given in Chapter V (Principles Of Pattern Cutting), beginning on page 79, which the reader will find a great help in explaining anything which he may fail to understand.