Let E I O B in Fig. 503 be the elevation of a pinnacle having four equal gables, down upon which a conical spire is required to be mitered, as shown.

Produce the sides of the spire until they meet in the apex D. Also continue the side E F downward to any convenient point below the junction between the spire and the gables, as shown by H, which point may be considered the base of a cone of which the spire is a part. Let V K L M be the plan of the gables The diagonal lines V L and M K represent the angles or valleys between the gables, while R S and T U represent the ridges of the gables over which the spire is to be fitted. Through the point H in the elevation draw a line to the center of the cone and at right angles to its axis, as shown by H C. This will represent the half diameter or radius of the cone at its base. With radius C H, and from center A2 of the plan, describe a circle, as shown, which will represent the plan of the cone at its base.

At any convenient distance from the elevation, and to one side, project a diagonal section corresponding to the line M A2 in the plan, as follows: From all the points in the side of the pinnacle draw horizontal lines indefinitely to the left, which will establish the hights of the corresponding points in the section. From any vertical line, as D1 A1, as a center line set off upon the horizontal lines the distances as measured upon the line M A2 of the plan. Thus make B1 A1 equal to M A2 and C1 H1 equal to A2 5, the radius of the cone at its base. The point F1 represents the hight of the crossing of the two ridges of the gables, therefore a line drawn from F1 to B1 will represent one of the valleys between the gables. Draw H1 D1, the side of the cone. Its intersection with the line of the valley at G will then represent the hight of the lowest points of the spire between the gables, and a line projected from this point back into the elevation, as shown, will locate those points in that view.

Fig. 504.-Pattern.

Fig. 503. - Plan, Elevation and Diagonal Section.

To describe the pattern, first divide one-eighth of the plan of the cone, choosing the one which miters with the gable shown in the elevation, into any number of equal spaces, as shown by the small figures. From these points carry lines vertically cutting the base of the cone H C, as shown, and thence toward the apex D, cutting the line B J of the gable, against which this part of the cone is to miter. As the true distance of any one of the points just obtained upon the line B J from the apex D can only be measured on a drawing when that point is shown in profile, proceed to drop these points horizontally to the profile line D H, where they are marked 11, 21, etc., and where their distances from D can be measured accurately. Next draw any straight line, as D' IF of Fig. 504, upon which set off all the distances upon the line D H of the elevation, all as shown, each point being lettered or numbered the same as in the elevation. With D2 of Fig. 504 as a center, from each of these points draw arcs indefinitely to the left, as shown. Upon the arc drawn from H2 set off spaces corresponding to those used in spacing the plan, beginning with H2, as shown by the small figures, and from each point draw a line toward the center D2 cutting arcs of corresponding number drawn from the line F2 H2. A line traced through the points of intersection (g to F2) will give the shape of the bottom of the cone to fit against the side of one of the gables, or one-eighth of the complete pattern.

By repeating the space 1 5 upon the are drawn from H2 seven times additional, as marked by the points 1 and 5, the point V will be reached, from which a line drawn to D2 will complete the envelope of the cone. From the points marked 1 and 5 draw lines toward D2 intersecting the arcs of corresponding number. This will locate all of the highest and lowest points of the pattern, after which the miter cut from g to F2 can be transferred by any convenient means, as shown from g to f, and so on, reversing it each time, as shown. In the case of a spire of very tall and slender proportions it will be sufficiently accurate for practical purposes to draw the lines g F2 and g f straight But the broader the cone becomes at its base the more curved will the line g F2 become. With a radius equal to D E of Fig, 503 describe the arc E2 I2, as shown, which will complete the pattern.