post. Join tliis point to c-e' and a templet for marking the brace is obtained.
In Fig. 7 a development of the two adjacent sides of the post is made, chiefly to show the limits within which the mortise and tenon must be made ; also how to mark the housing. The bevels are the same as those found for the brace, but a.re applied in reverse direction. The distance of point a from c is found in the plan ; the heights of the points are projected from Fig. 5.
Obtaining the Bevels of Splayed Linings—An opening fitted with a French casement frame, having splayed linings to jambs and soffit, is shown on page 149. One-half of plan and elevation is shown in detail, the other half in line diagram, showing the developments necessary to determine the true angles at the mitre, to obtain the bevels for marking the shoulders. The soffit is splayed at a less angle than the jambs, the usual method, as the splay is given only for effect, it is not necessary to obtain light ; the rays of light pass downwards not upwards, and the ceiling is lighted by reflection from the surfaces in the room. If the splays were alike, only one bevel would be required. The mitre line a' -b2, shown at top of Fig. 3, is the projection of the angle, and is not its real length or inclination. To obtain this we must turn the lining round parallel to the front. Let a, Fig. 1, be the pivot, move h, the outer edge of the lining, round to V \ the edges represented by a and b will then be in one plane. Referring to the elevation, point a' will not have moved in the operation, point b2 has moved out horizontally to b", which is the point where the two projectors meet. Join this point to a' and we have the shoulder line for the jamb. To obtain the bevel for the soffit. In like manner upon section Fig. 4, with a as centre and a -b3 as radius, describe an arc intersecting the vertical plane in bl. Project this point into the elevation, cutting the projector from the face of the jamb in b"".
Join this point to a' and the bevel for the head groove is disclosed.
The soffit would be laid upon the rod and point a' marked on each side, then the bevel just found applied to front edge with blade intersecting point a', and the line cut in. A second line about | in. farther out and parallel with the first, would be added to mark the groove for the reception of the tongue which is cut upon end of the jamb.
The bevel found for the jamb would be applied at the back to mark the shoulder, the tongue? being formeel on the face side, as shown in the plan.
Properties of and Methods of drawing Ellipses.— The ellipse is a figure used almost as frequently as the circle by the architectural elraughtsman. and its construction is const antly required in the workshop There is a considerable amount of misconception concerning this figure ; it is often confoundetl with other figures to which it has no relation, for instance the oval and the three-centred circular curve.
Definitions.—The Ellipse is a section of either a cone or a cylinder. It may be defined as a ¡Jane figure bounded by erne continuems curve described about two points (called the foci), so that the sum of the distances from any point in the curve to the two foci may be always the same (see Fig. 7, page 152).
Axes —A diameter of an ellipse is any straight line cutting it in halves by passing through its centre. One iliameter is conjugate to another when it is parallel to the tangents passing through the ends of the other. The' longest anel shortest conjugate diameters are at right angles with each other, and as the figure is symmetrical about them, they may be calle'd axes, and they are generally called the major (or greater) and mine>r (or lesser) axes respectively. The point of intersection of the two axes is calleel the centre of the ellipse, anel the axis of the generating cylinder (or cone») always passes through this point.
Ordinates are lines elrawn from the cire-umference perpendicular to the axis or diameter (see Figs, i anel 2).
Foci---Two points on the major axis, from which the curve lias a constant ratio—that is, the sum of the distance of any point in the curve from the two focal points is equal to the length of the major axis ; this will be demonstrated in reference to Fig. 7.
Normals are lines perpendicular to the curve at any particular point.
Tangents are lines at right angles to the normal and touching the curve.
Trammel—An apparatus or instrument for describing elliptic curves mechanically.
Trammelling .—A method of plotting an ellipse upon its axes similar in principle to the action of the trammel.
Methods of drawing the Ellipse.—There are many methods of doing this, several of which are shown on page 152. First as the section of a cylinder. Let a-b-c-d, Fig. 1, be the elevation of a cylinder, whose plan is shown in Fig. 2. To obtain the section made by a plane on the line A-A, divide the semi circumference of the plan into a number of equal parts, as 1, 2, 3, 4, 5, 6, 7, 8; project these points to the line of section as shown by the dotted projectors, numbering them similarly for easy reference. Erect perpendiculars to the line of section from these points, and make them equal in length to the corresponding ordinates in plan. Draw the curve through the points so found. Another plane; of section is shown by the line B-B, and the complete section is described on a line parallel to the line of section, hut clear of the generating solid. The same projectors are utilised as in the first case, the only difference in treatment being that the lengths of the ordinates are set otf on each side of the central 'ine or major axis, 0--8.
To Describe a Semi-Ellipse by means of intersecting lines, Fig. 3.—Draw a rectangle upon the major axis equal in height to the minor axis, as a -b, b-a. Divide each side into the same number of equal parts as shown ; join points a-1, 1-2, 2-3, 3-c, etc., and through the points where these lines intersect each other, draw the curve. If a sufficient number of divisions are used the curve will be described by
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