The first essential in beginning the work is an ellipse, which may be drawn by whatever rule is most convenient, and which must be of the length and breadth which the vase is required to have. Draw the plan of the sides of the vase about the curve, as shown in Fig. 373, in such a manner that all the points X, Y, Z, etc., shall have the same projection beyond the curve. Complete at least one-fourth of the plan by drawing miter lines, as shown by P C, M C, O C, U C and N C Above the plan construct an elevation of the article, as shown by H L K G. Only the profile H V W L of the elevation is needed for the purpose of pattern cutting, hut the other lines are desirable in process of designing, in order that the effect may be considered before the article is constructed. As the projection of each of the sides upon the plan is different when measured from the center C on lines at right angles to the lines of the sides, it will be necessary first to develop the profiles of each of the varying sides from the normal profile H V W L. Therefore, divide H V W L in the usual manner, and from the several points in it drop lines across its corresponding section (No. 1) of the plan.

Fig. S7S. - The Patterns for an Elliptical Vase Constructed in Twelve Pieces.

Across the second section in the plan, from the points already obtained in U C, draw lines parallel to O C, the side of it cutting O C, and produce them until they meet A C, which is drawn from C at right angles to C 0 produced. Then the points in A C give the projections from which to obtain a profile of the section numbered 2. In like manner continue the points from C O across the third section in the plan, parallel to O M, the side of it cutting M C, and produce them until they cut C B. which is drawn from C at right angles to O M produced. Then C B contains the points requisite in obtaining a profile of the third section. Continue the points in C M across the fourth section, cutting its other miter line C P. From C draw C D at right angles to the side P M of the section, cutting the lines drawn across section 4. Then upon C D will be found the points necessary to determine the profile of the fourth pattern. Produce the line of the base of the elevation indefinitely, as shown by C1 C2 C3, and also the line of the top A' B2 D3 From the several points in the profile H V W L draw lines indefinitely, parallel to the lines just described and as shown in the diagram. From C1, upon the base line produced, set off points corresponding to the points in C A of the plan, making the distance from C in each instance the same as the distance from C in the plan. Number the points to correspond with the numbers given to the points in the profile H V W L, from which they were derived. In like manner from C2 set off points corresponding to the points in C B of the plan, numbering them as above described. From C3 set off points corresponding to those in CD of the plan, likewise identifying them by figures in order to facilitate the next operation. From C1 erect the perpendicular C1 A1; likewise from C2 and C3 erect the perpendiculars C3 B2 and C3 D3. From each of the points laid off from C, and also from each of those laid off from C2 and C3, erect a perpendicular, producing it until it meets the horizontal line drawn from the profile H V W L of corresponding number. Then lines traced through these several intersections will complete the profiles, as shown. Perpendicular to the side of each section in the plan lay off a stretchout taken from the profile corresponding to it, just described, and through the points in the stretchouts draw measuring lines in the usual manner, all as shown by E F, E1 F1, E2 F2 and E3 F3. Place the T-square parallel to each of these stretchout lines in turn, and, bringing it against the several points in the miter lines bounding the sections of the plan to which they correspond, cut the measuring lines in the usual manner. Then lines traced through the points of intersection thus obtained, all as shown in the diagram, will complete the patterns.