The hypothenuses of the various triangles will thus represent the true distances across the pattern upon the solid lines of the plan, while the distances upon the dotted lines can be represented by pieces of thread or wire, placed so that each will reach from the point at the base of one of the triangles to the point at the top of the one next it. If constructed of metal two or three triangles will suffice to give the model sufficient rigidity, and the remaining points can be connected by pieces of wire, using a different kind of wire to represent the distances on the dotted lines.

In Fig. 264, is shown a pictorial representation of a model constructed, as above described, from the drawings shown in Fig. 261. In the illustration the triangles 2, 5 and 8 only are shown in position, their hypothenuses connecting points of similar number in the upper and lower bases. The other points are represented as being connected by wires or threads repre-senting both the solid and the dotted hypothenuses in the diagrams of triangles in Fig. 261. Such a model if constructed will give a general idea of the shape of the entire covering, and at the same time of the small pieces, or triangles, of which the covering is composed, with all the dimensions of each. If all of the spaces formed upon this skeleton surface could be filled in with pieces of cardboard or metal just the size of each and the whole removed together and flattened out (each piece being fastened to its neighbor at the sides), it would constitute the required pattern, the same as will be subsequently obtained by measurements taken from the drawing, and as shown in Fig. 265.

Fig. 264. - Perspective View of Cardboard Model of One-half the Article Shown in Fig. 261.

Having by means of the diagrams of triangles in Fig. 261 obtained the Lengths of all the sides it is now only necessary to construct successively each triangle in the manner described in Chapter IV (Geometrical Problems), Problem 86, remembering that the last long side of each triangle used is also the first long side of the next one to be constructed. Therefore, at any convenient place draw-any straight line, A N of Fig. 265, which make equal to the real distance from A to N, Fig. 261, which has been found to be the distance 0 0 of the diagram of solid lines. To conduct this operation with the greatest economy and ease it is necessary to have two pairs of dividers, which shall remain set, one to the spaces upon the plan of the base ABC, and the other to the spaces upon N" O" P", and a third pair for use in taking varying measurements. From A of Fig. 265 as a center, with a radius equal to 0 1 of the plan, Fig. 261, describe a small arc, and from N as a center, with a radius equal to the true distance from N to 1 of the plan, which has been found to be 0 1 of the diagram of dotted lines, describe another arc, cutting the first one as shown at the point 1, Fig. 265. The triangle thus constructed represents the true dimensions of one indicated by the same figures of the plan. Next from N of the pattern as a center, with a radius equal to N" 1 of true profile of top, Fig. 261, describe a small arc, which cut with one struck from point 1 of pattern as a center, with a radius equal to 1 1 of the diagram of solid lines, thus locating point 1' of pattern. This triangle is, in turn, succeeded by another whose sides are next in numerical order, that is 1 2 of the base and 1 2 of the diagram of dotted lines. Thus the operation is continued, always letting the spaces of the circumference of base succeed one another at one side of the pattern, and the spaces upon the true profile of top succeed one another at the other side of the pattern, until all the triangles have been laid out as shown by A N P C, Fig. 265, which will complete one-half the entire pattern.

It is not necessary to draw all of the dotted or solid lines across the pattern, as the points where the small arcs intersect are all that are really needed in obtaining the outlines of the pattern, but it is often advisable to draw them as well as to number each new point as obtained, in order to avoid confusion and insure the order of succession.

In dividing the curves of top and bottom into spaces, such a number of points should be taken as will insure the greatest accuracy, as in the case of dividing a profile. Thus too few would give too short a stretch-out, while if the spaces were too small error in transferring their lengths might result, which would be increased as many times as there were spaces.

Under the head of transition pieces may be included a large number of forms having various shaped polygonal or curved figures as their upper and lower surfaces, placed at various angles to each other, sometimes centrally located as they appear upon the plan and sometimes otherwise. It often happens that one surface or termination is entirely outside the other in that view, forming an offset between pipes of differing sizes and shapes. Sometimes such an offset takes a curved form, constituting a curved elbow of varying section throughout its length, in which case it consists of a number of pieces, each with a different shape at either end. With such forms may be classed the ship ventilator, whose lower end is usually round and horizontal and whose upper end is enlarged and elliptical and stands in a vertical position, the whole being com-' posed of five or six pieces. In such east's, when the shape and position of the two terminating surfaces only are given, it becomes necessary to assume or draw as many intermediate surfaces as there are joints required, each of such a shape that the whole series will form a suitable transition between two extreme shapes. It may be remarked, that what have been spoken of here as "surfaces" do not necessarily mean surfaces of metal forming solid ends to the pieces described, but simply outlines upon paper to work to, as more often the "surface" is really an opening.