In Fig. 592, ABCD shows the elevation of the article, below which E F G H shows the plan at the bottom and E J K L the plan of its top, both circles being tangent at the point E.

Divide the circle representing the plan of the top into any convenient number of equal spaces, as represented by the small figures between KLE in the diagram. In the illustration only one-half of the plan has been divided, which is sufficient for the purpose. Next divide a like portion of the plan of the base into the same number of equal parts, as shown by the figures between E H G. Connect these two sets of points, first by lines drawn between like numbers, as 1 and 1'. 2 and 2', 3 and 3', etc. In a like manner connect 1 of the inner circle with 2' of the base, 2 with 3', 3 with 4', etc., all as shown by the dotted lines in the plan. These lines just drawn are the bases of a number of right-angled triangles, whose altitudes are equal to the vertical hight of the article, and whose hypothenuses, when obtained, will give the correct measurements across the pattern between the numbered points.

For a diagram of triangles representing the solid lines in plan erect the vertical line P S in Fig. 593, equal to A B of elevation. Then at right angles from S lay off a base line, upon which set off distances, measuring from S, equal to the lengths of the several solid lines drawn across the plan in Fig. 592. Thus make S R equal to K G (1 1') and S 2 to 2 2', and so on. From the points thus established in the base draw lines to the apex. Then the hypothenuses of the triangles will be equal to measurements on the surface of the finished article on lines drawn from the points in the base to corresponding points in the top. In the same way construct the diagram representing the triangles based on the dotted lines in plan, as shown in Fig. 594. Set off T V equal to the straight night of the article. From V draw the horizontal line V U, upon which, measuring from V, set off distances equal to the length of the dotted lines across the plan. Thus make V 2 equal to 1 2', V 3 equal to 2 3', etc. From the points thus established in V U draw lines to the apex T. These lines will be equal to measurements upon the surface of the finished article between the points connected by the dotted lines in the plan.

Fig. 592. - Plan and Elevation of Flaring Article, Showing Method of Triangulation.

Fig. 593. - Diagram of Triangles Based upon Solid Lines in the Plan in Fig. 592.

Fig. 594. - Diagram of Triangles Based Upon Dotted Lines in the Plan in Fig. 592.

Having obtained the correct dimensions of all the triangles assumed at the beginning of the work they may now be constructed consecutively, thus developing the pattern in the following manner: Assume any straight line, as D C of Fig. 595, which make equal to D C of the elevation, or, what is the same thing, P R of the diagram of triangles, Fig. 593. From C as a center, with a radius equal to 1' 2' of the plan, strike a small arc, which intersect with another small arc struck from D as center, with a radius equal to T 2 of Fig. 594, thus establishing the point 2' of the pattern. From D as center, with a radius equal to 1 2 of the plan of the top, Fig. 592, strike a small arc and intersect it with another struck from 2' of the pattern as center, and P 2 of Fig. 593 as a radius, thus establishing the point 2 in the top of the pattern. Proceed in this manner, using the hypothenuses of the triangles in Fig. 594 with the spaces in the Outer curve of the plan, Fig. 592, to establish the points in the bottom curve of the pattern; and the hypothenuses of the triangles in Fig. 593 with the spaces in the inner curve of the plan to establish the points in the top curve of the pattern. Lines traced through the points of intersection, as shown from C to B and from P to A, will, with D C and A B, constitute the pattern for the half of the article shown by E H G K of the plan. The other half may be added, as shown, by any convenient means of duplication.

Fig. 595. - The Pattern of Flaring Article Shown in Fig. 592.

Since the top and the bottom of this figure are both round and horizontal, it becomes the frustum of a scalene cone, which permits of its being treated by a different, and perhaps simpler, method of triangulation, all of which is given in Problem 167, to which the reader is referred.

PROBLEM 185. - The Pattern of an Article having: an Elliptical Base and a Round Top.

Fig. 596 shows the plan and elevation of the article for which the pattern is required. Divide one-quarter part of the plan of the base E G into any convenient number of equal spaces, and divide a corresponding part of the plan of the top L K into the same number of spaces, numbering the points of division the same in both, as indicated by the small figures 1, 2, 3 and l1, 21, 31, etc. The article here shown possesses some of the general features of the cone in that it is tapering in its sides, but inspection will show that the slant or taper of its sides varies in different parts of its circumference, or in other words, that different lines drawn through like numbers in the base and top would, if extended upward, meet the axis at different hights, hence some means must be devised for measuring the real distances between the points in the base and the points in the top, which may be accomplished in the following manner: First connect all points in the base in plan with points of the same number in the top by means of a solid line, as shown upon the plan by lines 1, 1', 2, 2', etc. Also draw the intermediate dotted lines connecting alternate points, as shown in the engraving by 2 l1, 3 21 4 31, etc., thus dividing the entire surface of the article into triangles. Construct a diagram, as shown by A1 N1 C1, Fig. 597, in which the actual distance between corresponding points in base and top shall be shown. Make C1 N1 equal to the straight hight of the article, C N of the elevation. At right angles to it set off N1 A1, in length equal to the distance 11 1 in plan. From V set off also on N1 A1 spaces corresponding to 21 2, 31 3, 41 4, etc., of the plan, and from each of these points draw a line to C, as shown. Then the lines converging at C1 represent the distances which would be obtained by measurements made at corresponding points upon the article itself. Construct a like diagram of the distances represented in the dotted lines in the plan, as shown by C2 N2 O, Fig. 59S. Make C2 N2 equal to C N of the elevation, and from N2 set off at right angles the line N2 O. Upon this line make the spaces N2 2, N2 3, N2 4, etc., equal to the length of the dotted lines 11 2, 21 3, 31 4, etc., and from the points thus obtained in N2 O draw lines to C2. Then these converging lines represent the same distances as would be obtained if measurements were made between corresponding points upon the completed article.

Fig. 596. - Elevation and Plan of Flaring Article with Elliptical Base and Round Top.

Fig. 597. - Diagram of Triangles Based upon Solid Lines of the Plan in Fig. 596.

Fig. 598. - Diagram of Triangles Based upon Dotted Lines of the Plan in Fig. 596.

Fig. 599. - The Pattern of Article Shown in Fig. 596,

For the pattern, commence by drawing any line, as P X in Fig. 599, on which set off a distance equal to C1 l of the first diagram, as shown by 1 l1. Then, with the distance from 1 to 2 of the plan for radius, and 1 in pattern as center, describe an arc, which interseet by another arc struck from 11 of the pattern as center, and C2 2 of the second diagram as radius, thus establishing the point marked 2 in the pattern. Next, with 11 21 of the plan as radius, and from l1 of the pattern as center, describe an arc, which intersect by another are drawn from 2 as center, and with C1 2 of the first diagram as radius, thus locating the point 21 of the pattern. Continue in this manner, locating each of the several points shown from X to Y and from P to R of the pattern, through the several intersections, tracing the lines of the pattern, as shown. Then X Y R P will be one-quarter of the required pattern. Repeat this piece three times additional, as shown by W X P T, V W T U and Y Z S R, reversing each alternate piece, thus completing the pattern.