and a perpendicular dropped from i and 5. These points are also the centres of the quadrants containing the lower centres, anil the remainder of the construction is clearly indicated in the. figure. On the left the span is divided into four equal parts, and an equilateral triangle constructed upon the middle, two give the centres as shown.
The Cyina Reversa or Wave Arch, Fig. 6, is produced upon a triangle of 6o°, the span ct--h forming the base. Bisect the inclined sides a-c and b-c in points e and g, and draw a line through the points parallel with a-b. With c and g as centres and the length e-g as radius, describe the arcs e-c and g-c. With the same radius, describe arcs from a and b, intersecting the line of centres in d and /. These locate the two centres for the remaining arcs of the curve.
The Ogee Arch. Fig. 7, is a form frequently adopted for pavilion roofs, also for window frames. To describe it, construct an equilateral triangle on the span. From the three angles of the triangle with radius equal half the span, describe arcs intersecting the sides as shown at the points of intersection ; describe other arcs intersecting the first, in points 1, 2, 3. These points furnish the centres for describing the curves. A line joining points 1-2 or 1-3 gives the position and direction of the joints.
The Bell Arch or Reversed Ogee, Fig. 8, is drawn similarly. Join the crown to the springing by the line c-s. Bisect this line, which gives the point of junction of the curves of contrary flexure. With a radius equal to half the length of c-s, describe intersecting arcs above and below the line. These intersections give the centres of the required arcs. If a joint is required it must be upon a line joining the centres of the reverse arcs.
Compound Curves.—There are many of these, and a few that are considered of most service to artisans in the building trades are dealt with.
The Helix. Figs. 1 and 2, page 173, are the plan and elevation respectively of a cyh'ndric helix, sometimes miscalled a spiral. The spiral is a curve of continually
diminishing radius, no two portions being alike. The helix is a curve of constant distance from its axis of revolu tions, its opposite portions being symmetrical. The plan of a helix may be a circle or an ellipse ; the elevation is similar in each case. The projection of a helix as shown in Fig. 2 does not give its true contour, because the curve itself is formed upon a cylindric surface, whilst the projection is upon a plane ; but the projections are necessary for many purposes, one of which is shown in Figs. 4 and 5. Fig. 3 is the; development of the helical curve shown in Fig. 2, L-c being the helix, which, it will be observed, is a straight line. a-b represents the distance w hich one revolution of the helix rises or adv ances, and this is called the " pitch," a term which is not strictly correct; the distance a-b upon either iigure gives the amount ot pitch or inclination of the curve, but it is not the inclination itself. In the example, one in. (by scale) is taken for each complete revolution of the line ; therefore such a helix is termed one of an inch pitch. A common example is the thread of a screw. The distance between the highest part of a thread and the highest part of the next one above it, upon the same side, gives the " pitch " of that screw, otherwise expressed, as so many threads to the inch.
To draw the Helix.—-Describe a circle of the required diameter; divide this into a number of equal parts, as shown in Fig. 1. Draw a line parallel with one diameter, and erect a perpendicular, as a-B. Set off upon this the desired rise or amount of " pitch " for one revolution, as a-A. Divide this space into the same number ot equal parts as the plan, and number them similarly. Next draw projectors from each point upon the circumfercnce of the circle into the elevation, and intersect them by horizontal projectors dra wn from the correspondingly numbered divisions between a-A. Trace the curve through the points so lound. If desired, intermediate points may be used to facilitate the freehand drawing, as shown by the dotted line between the points 2-3 and its projector, shown in full lines. Note that at each revolution the curvc repeats upon the projection.
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