1470

THEORY AND PRACTICE OF ENGINEERING.

Book II.

The hollow AabcdefTZ is the primitive centre of the vault; the easiest method ofconstructing such a stair is first to set out that part of the hollow cylinder, ax my,in which each voussoir is comprised, according to its projection mm,nn , on the plan,in order to trace on their curved surfaces their ramp and arrises, as well as the heights andwidths of the steps to which they correspond. In regular staircases like the present,the steps being equally divided, give for the developments of the concentric circles bdb ,right lines more or less inclined, which may be drawn on the curved surfaces corre-sponding to them with pliable rules. This operation being performed for the arrisesmm,ran, of the thickest voussoirs, we may afterwards cut away the rampant partand form the smooth soffite and extrados. To trace the voussoir, apply at each endthe mould medn of the primitive centre: the curve of the intrados and joints being setout, draw on the intrados two lines parallel to the curved edges of the smooth soffite;then to form these joints cut away the stone beyond the lines, applying the rulealways vertically from one curve to the other, and hollow the intrados according to thecurves traced at the two ends, by means of compasses, taking care to keep perpendicularto the side on which is the curve; the voussoir is terminated by cutting the two ends n3and & 4 at right angles to the extrados.

Another method to perform this Operation , and to economise Stone . — Commence bymaking the horizontal projection mm,nn, of the voussoirs X, proposed to be executed, anddraw the geometrical elevation in order ; to find the section of an oblique cylinder in whichit may be comprised, indicated by the lines mn and ba , representing two parallel planes,which pass through the highest points, m,n, and the lowest points, a, b, of the voussoir whenin its place. To have the inner curve, which produces this section, divide the chord nnof the arc, expressed in the horizontal projection, into equal parts, through which drawthe ordinates of the curve; then divide the inclined chord nn into the same numberof equal parts, through which raise other perpendiculars, and set off on each theheight of that which corresponds to it in the horizontal plane. This effected for thetwo curves will give the mould mmnn, for the section of the oblique cylinders inwhich the voussoir should be comprised, and which may be formed with a stone, whosethickness will be equal to the distance between the oblique lines mm and ba, prand qs. This stone should have a face at right angles with the two parallel surfaces,answering to the chord nn of the inner curve, for the purpose of tracing from the elevationthe vertical lines nb,nb , which serve to fix the position of the mould, mmnn , withwhich the contours are traced, to form the oblique section of the cylinder in which thevoussoir is comprised: this section being made on the inner and exterior curved sur-faces, draw the ramp lines in proportion to their heights 56,n7, and the breadths n6,57, of the steps to which they correspond, proceeding as in the preceding method for theramps, the joints, and the extremities of the voussoirs, which should be perpendicular tothe lines of ramp.

Vis Saint GiUes , on a square Plan. — This example is composed of jointed vaults, andskewed as well as rampant arc de cloitre ; its execution is attended with some difficulty.The centres corresponding to the arrises of the steps, perpendicular to the middle of thefaces, can alone be right lines, all the others being oblique, and forming more or less elongatedellipses, according to the steps to which they appertain. The plan of the vault shows onehalf the arrangement: the primitive centre is a semicircle IKL, raised perpendicular onI L as a diameter ; on this semicircle the voussoirs are set out through the points a, b, c, d, e,fthrough which lines parallel to the faces B C and F G of the wall are drawn, till theyintersect the diagonals FB, GC; these lines, which indicate the projections of the rows ofvoussoirs, would form on the plan of the entire vault squares inscribed in each other, as inthe projection of an arc de cloitre; the two halves of the primitive centre are representedon the section, with the thickness of the vault, extradossed in a right line, and in thedirection of the steps it is to sustain ; this figure is a section of the rampant part of thevault, forming an arc de cloitre along the wall B C, and of the portion returned, C D, withthe projection of the joints corresponding to those indicated on the plan by aa,bb y cc.Half the section is shown on the side of the newel, forming an arris vault, withthe direction of the joints corresponding t off,ee,dd, on the plan. To trace the voussoirscommence by drawing on the plan and elevation the particular voussoir to be executed,in order to form the mould, which comprises its greatest length, breadth, and height; havingthen cut a prism, whose base is the plan of the voussoir, trace on its vertical surfacescurved or right lines with the moulds taken from the corresponding section.

To draw the rampant vault forming the joint gh , divide the arc a b of the primitivecentres into equal parts, and draw through these points lines cutting gh , which maybe regarded as the profile of a vertical plane passing through this joint; the inter-sections give the heights, which, starting from the point h, correspond to the projec-tions shown on the plan by parallels drawn from the division points of the primitive arcab on the projection of the same joint, indicated on the plan by gh, where this line repre-