In Fig. 620 G E F shows the plan of the article at the base, L J K the plan at the top and ABCD an elevation of one side. An inspection of the plan will show that the article consists of two symmetrical halves when divided by the line G H, and that, therefore, the triangulation of one-half will answer for the whole. On account of the dissimilarity between the outlines of the top and the bottom some judgment will be required in adopting a good division of the surface into triangles.

Fig. 620. - Plan and Elevation of an Article Whose Top is a Circle and Whose Base is a Quadrant.

Fig. 621. - Diagram of Triangles.

As the point L of the plan is the nearest point to the adjacent side E G, it must be chosen as the vertex of a triangle whose base is E G. That portion of the circle of the top, therefore, between L and its corresponding point K in the other half of the article must be considered as the base of an oblique cone whose apex is at G.

It is always advisable in the division of a surface into triangles that the solid and dotted lines crossing the plan should intersect the outlines of the top and the bottom as nearly at right angles as possible.

Therefore, since the remainder of the top (L to J) and

E H of the base are the liases of a surface which must be so divided as to best serve the purposes of triangulation, it is advisable to divide L J into more spaces than K H, allowing the extra spaces in the top nearest the point L to form the bases of a number of converging triangles, as shown. Thus first divide E H into any suitable number of spaces, as shown by the small figures 5 to 9, then divide L J into a greater number of equal spaces than E H, as shown by the small figures 3 to 9. Connect points of like number in the two outlines by solid lines, commencing at H and J, as shown from 9 9 to 5 5, drawing lines also from 4 and 3 of the top to 5 (E) of the bottom. Also draw the dotted lines 5 6, 6 7, etc., and the solid lines from points in L N to G. These solid and dotted lines will then form the liases of a series of right angled triangles whose hypothenuses will give the real distances across the envelope of the finished article.

These triangles are constructed, as shown in Fig. 621 at the right of the elevation, in the following manner. Extend A B and D C of the elevation, through which draw any vertical line, as Q R. From Q on Q P set off the lengths of all the solid lines of the plan. Thus make Q 9 equal to 9 9 or J H of the plan, Q 8 equal to 8 8 of the plan, etc., and from the points thus established draw lines to E. In like manner draw the vertical line T V, and from T on T S set off the lengths of the dotted lines of the plan, as shown by the small figures, and from the points thus obtained draw lines to V, as shown. The small figures in S T correspond I with the figures in L J, the top line of the plan.

In laving out the pattern shown in Fig. 622 the joint is assumed upon the line J H of the plan. The pattern may be best begun by first laying out one of the large triangles forming a side of the article, as E L G or G K F of the plan, shown also by D N C of the elevation, Fig. 620. Draw any horizontal line, as E G of Fig. 622, equal in length to E G of the plan.

From E as center, with radius R 3 of Fig. 621, describe a small are near L, which intersect with another arc drawn from G as center, with a radius equal to R 3' of Pig. 621, thus establishing the point L of the pattern. From G of the pattern as center, with radii equal to R 2 and R 1 of Fig. 621, describe small ares, as shown between L and K of the pattern. Take between the points of the dividers a space equal that used in dividing the are L K of the plan, and placing one foot of the dividers at L of the pattern step from arc to arc, reaching K, as shown, and through the points thus obtained draw L K of the pattern; also draw K G. From E of the pattern as center , with radii equal to R 4 and R 5 of Fig. 621, describe small arcs to the left of L. With the dividers set to the space used in dividing the are L J of the plan, place one foot at L (3) and step first to are 4. then to arc 5, thus establishing the points 4 and 5.

Fig. 622. - Pattern of Article Shown in Fig. 620,

With the last obtained point, 5, of the pattern as center, and a radius equal to the dotted line V 5 of Fig. 621, describe a small arc (6'), which intersect with another arc struck from point E of pattern as center, with a radius equal to 5 6 of the base line E H of the plan, thus establishing the point 6' of the pattern. With 6' of the pattern as center, and a radius equal to R 5 of Fig. 621, describe a small arc (6), which intersect with another arc struck from 5 of pattern as center, with a radius equal to 5 6 of the top line L J of the plan, thus establishing the point 6 of the pattern. Proceed in this manner in the construction of the remaining triangles of the pattern, using alternately the lengths of the dotted and the solid hypothenuses in Fig. 621 corresponding to the dotted and the solid lines crossing the plan, in the order in which they occur, to determine the width of the pattern; the spaces in E H of the plan to form the lower line E H of the pattern and the spaces in L J of the plan to form the upper line of the pattern, all as shown. The remaining parts of the pattern can be obtained by any convenient means of duplication, K F G being a duplicate of LEG and K J1 H1 F being a duplicate of L J II E.