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Spiral lines & Nave and Transept Roof

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the vertices of the polygon ; then with A, B as the given length step off with the dividers the remainder of the sides, as on Figs. 3 and 4; the method is the same in all regular polygons. A, B re-presents the horizon and the second space from C the number of degrees or distance the diagonal line rises.

Fig. 6 shows the method of describing a circle through any three given points out of a right line, or to find the center of an arc. Let A, B, C be the given points; with any radius and A, B, C as centers, describe the arcs 1, 2, 3 and 4. Draw the lines through their points of intersection; then at the intersection of the lines will be found the center from which to describe the circle touching the points A, B, C.

Fig. 7 shows the method of describing any number of spiral lines around a given center whose diameters are all alike on the horizon. Let 1 be the center or given point; divide the horizon into twice the number of parts there are to be revolutions of the line ; then with 1 as a center describe all the arcs above the horizon; and with 2 as a center, describe all the arcs below the horizon and the spiral is complete.

Fig. 8 shows the method of obtaining the circumference of a circle geometrically; with the diameter A, B as a radius, describe the arcs intersecting at C. Draw the tangent D, E parallel to A, B, indefinitely, draw the lines C, A, C, B to intersect the tangent at D, E; then the distance from D to E will be equal to one-half the circumference of the circle.

Fig. 9 shows the method of describing any number of spiral lines around a given center where lines diverge one or more diameters on the horizon for every revolution of the line; divide the horizon into twice the number of parts there are to be revolutions of the line; with one point of the dividers at X de-scribe the first diameter, then with 2, 4, 6, 8 as centers, describe successively the arcs above the horizon and in the same manner describe the arcs below the horizon from 1, 3, 5, 7 on the opposite side of the center, changing the radius and center for each semicircle.

Fig. 10 shows the method of describing an ellipse by means of a cord; make A, C, B, C each equal to one-half the longest diameter 1, 2. Make C, D equal one-half the shortest diameter ; drive a nail or pin at A, B, C, around which points tie the cord ; then remove the pin at C, and, with the pencil inside the cord, describe the figure as shown, taking care to keep the tension of the cord the same at all points.

Fig. 11 shows the method of describing an arc of large dimension by means of a triangle. Let A, B, C be the points through which to describe the arc; the line A, C will then be the cord of the arc. Out of suitable material construct a triangle, making E, F parallel to A, C, and twice the length of the arc at B, D; at each end of the cord A, C place a nail or pin, with the pencil at the vertex B, move the triangle from A to C, keeping it tight against the pins, and the arc is described.

Fig. 12 shows the method of describing an equilateral triangle from a given circle. Take any diameter, A, B, with half the diameter as a radius and B as a center describe arc E, F, connect E, F, F, A and A, E, and A, E, F will be the points of the triangle.

PLATE III.
Fig. 1 shows the plan of a nave and transept roof with hip, valleys and gables.
Let A, B show the run of the hip rafter, and B, D the run of the common rafter; B, E or B, C will be the rise of either of them, as both are the same. Now join A, E, which will be the length of the hip, and D, C the length of the common rafter ; the bevel shown at A will be for the foot of the hip and the one at E for top or down bevel. The bevel at C will be the down bevel for the common rafter and the one at D for the foot or horizontal bevel ; now drop the length of the hip, A, E, to intersect the line of the deck, on the line of the plate, A, D, space off the jack-rafters, extend the lines to intersect the line of the hip, which will be the length of the jack-rafters.

The bevel shown at the point of intersection will be the face or angle bevel for the top of the jacks; the down and foot bevels being the same as the common rafter. On the valley, 3, 3, is shown the same method of getting the lengths and bevels of the valleys, except the bevels on the jack-rafters reverse.

At 5 on the plan is shown the method of getting the angle to back the hip rafter so that when in position it will be upon the same plane as the common rafter. Space off equal distances from the corner 6-9, 6—10; with 10 as a center describe the arc shown, making it tangential to the common rafter, 6, 7, make 9, 11 tangential to the arc ; draw the diagonal line, 6, 12, connect 10 with the point of intersection of 9, 11 and 6, 12. The angle shown at 5 will be the backing of the hip.

On Fig. 2 is shown a sectional view of the hip and the jack rafter when in position and the manner of laying them off. Many carpenters, after getting the lengths as shown on Fig. 1, proceed to lay off their work, making their