For greater accuracy in the case of a very tapering cone, the circles of the. plan can be completed, as shown dotted, and their points of intersections with the line Z N can be dropped into oblique elevation, as seen at S and T, through which a line can be drawn to meet a line through F and H with greater accuracy than one through Y' and X', as the angle in the former case is twice as great.

In the above methods of obtaining the envelopes of what may be termed irregular conical forms, it will be clearly seen that the operation of dividing the curve of the base into a great number of spaces really re solves the conical figure into a many sided pyramid, and that the lines connecting the apex with the points in the base, which have been referred to as hypothe-nuses, are really the angles or hips of the pyramid. It is therefore self evident that any method of development which is applicable to a many sided pyramid is equally applicable to one whose sides are fewer in number, with the only difference, however, that the lines representing the angles or hips in the case of a pyramidal figure mean angles or sharp bends in the pattern of the envelope, while in the case of the conical envelope the bends are so slight as to mean only a continuous form or curve.

It is believed that the foregoing elucidation of the principles governing the development of the surfaces i of irregular shaped figures is sufficiently clear to make the demonstrations of this class of problems, given in Chap. VI, Section 3, easily understood by the student, as well as to enable him to apply them to any new forms that may present themselves for solution.

This chapter is intended to present, under its three different heads, all the principles necessary to guide the student in the solution of any problem that may arise. Its aim is to teach principles rather than rules, and the student is to be cautioned against arbitrary rules and methods for which he cannot clearly understand the reason. His good sense must govern him in the employment of principles and in the choice of methods. There is hardly a pattern to be cut which cannot be obtained in more than one way. Under some conditions one method is best, and under other conditions another, and careful thought before the drawing is begun will show which is best for the purpose in hand.

Fig. 271. - Elevations and Plan of an Article the Corner of which is a Portion of a Scalene Cone.

Principles of Pattern Catting.

The list of problems and demonstrations in the chapter which follows is believed to be so comprehensive that therein will be found a parallel to almost anything that may be required of the pattern cutter, and it is believed that he will have no difficulty in applying them to his wants.