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Problem I.— Given one of the ribs of the transverse ranges,*°find the body-range.

Suppose the given rib to be a semi-circle, or a semi-ellipsis,“escribe a semi-ellipsis, the length of which is that of thebody-range, and the semi-conjugate axis the height of the; S'Ven rib, which is that of the groin ; and the semi-ellipsisttius described will be the contour of the ribs which are to‘°rm the body-range.

In the same manner, if a rib of the body-range be given,‘hat of the transverse openings may be found.

, But if the given rib be any other curve than that of a semi-Cl fcle or semi-ellipsis, lay down the curve of half the given ribhpon any straight line, as a base; from the point where thec hrve intersects the base, draw another straight line at a righta hgle therewith : upon this line set the extent of half the^idth of the range from the angular point complete theRectangle , of which these two straight lines are adjoiningSl des ; draw that diagonal of the rectangle which meets thebUrve given ; take any number of points in the given curve,“raw ordinates perpendicular to the base to intersect them ;r°tn these points draw lines parallel to the other two sides of. e rectangle, to meet the diagonal; from the points of divi-J ,0n » draw straight lines parallel to the base of the given rib,b meet the side of the rectangle, which joins the curve of theS’ven rib ; from these points raise perpendiculars ; transferj>e ordinates of the given rib upon the perpendiculars ; andbe curve drawn through the extremities of these perpen-lc ulars will be that of the rib required.figure 1, No. A,fqk gf is the given rib, gf its base, equal0 half the width, c d, No. 3, of the transverse openings ;^ 1 e f is the rib found from the given rib f qkgf, and f e its4se , equal to half the width, a b, of the body-range.

„ figure 1, No. 5. Shows the method of finding the mould? r drawing the angles for placing the jack-ribs. The opera-l ° n is thus performed :

figure 1, No. 5. Upon any straight line, m l, transfer theJ^fti, No. 4, stretched out with all its parts from l tom;

the points thus transferred, erect perpendiculars, and“ a nsfer e h, No. 4, to the middle perpendicular, n o, No. 5 ;ben the remaining parallels of e h of the triangle e h f No. 4,bspectively to the parallels of n o, No. 5, on each side of it;jj d through the remote extremities of these perpendiculars,a curve each way from the point o ; and thus two equalbrves will be formed. But lest the reader should not bej® to follow a general description, the following shows notJlv how any particular point may be found in the requiredb> but also how any point may be found in the covering,■b.h'st: To find a point in the required rib.

^ figure 1, No. 4. Take any point, q, in the given rib;^ a W qp perpendicular to gf the base; draw p r parallel^7e, cutting fh at r; draw rs parallel to gf cutting/^

. *s draw st perpendicular to fe; make s t equal to pq,

^ d < is a point in the curve. In like manner, as many pointsY he found as required.

^■Now let it be required to find the point u, No. 5. Transferarc it, No. 4, upon the straight line ml, No. 5, from^ b u s draw uv perpendicular to ml; make mu equal to sr,jd v is a point in the curve. The trilinear area mo l is the| e l°pe of the portion of the groined surface represented byIj ber of the two triangles of the plan of the groin, which^ Ve the width of the body-range for the base; mo is theof a mould by which the angle of the groin, or the^ .'bdritic lines, are found ; the point m is laid to the bottom,

. the mould bent upon the surface, so that the point o mayb jhe summit, and the convex side of the mould towards(brtical section passing through the point from which theto m of the mould rises.

Figure 1, No. 6. Shows the envelope corresponding toeither of the two triangular parts of the plan of the transverseparts, and is found in the same manner as No. 5. Thisshows the mould for cutting the boarding, by laying it outto the full breadth upon a plane, and drawing the ends of theboards by the curved edge of the mould, so as to fit againstthe boarding of the body-range.

Figure 2. Shows the construction of a groin upon aninclined plane, the widest opening, or body-range, having itsascent or descent in the direction of the inclination of theplane; the transverse ranges are therefore level. The ribsin both directions are set in vertical planes. The rib for thebody-range is a semi-ellipsis ; those of the sides will also besemi-ellipses, but will not have their axes in a vertical andhorizontal position. The elevation of the transverse openingsis shown at No. 2, and that of the body-range at the lowerend. Each elevation exhibits the construction of the centre,a section of the boarding, and the manner of placing thejack-ribs.

No. 3. Is a section of the body-range at right angles tothe plane of its inclination.

No. 4. Shows the form of the moulds for drawing theangles, in order to place the jack-ribs. The construction isthus : Divide the half, df of the curve d e, No. 3, into anynumber of equal parts ; from the points of division drawlines parallel to the axis of the body-range, cutting thediagonal AL; from the points of intersection draw linesparallel to the sides of the transverse opening, and continuethem on the other side of the inclination, g h, of the elevationof the said openings ; from g h, as a base, make the heights ofthe several lines thus continued equal to the correspondingheights of the elevation of the body-range, and a curve drawnthrough all the extremities will give the form of the ribs, ip k,for the transverse openings. From the summit of the centrethus constructed, draw the line at right angles to theinclination gh; extend the arc df No. 3, with its divisions,to vw, No. 4; through all the points of division in v w drawlines at right angles thereto ; from all the points found in thecurve of the rib ip k, No. 2, draw lines parallel to i h, cuttingthe respective lines perpendicular to v w; and a curve drawnon each side of v w will give the angles of the groin.

Groin Ceiling, a cradling constructed of ribs, lathed andplastered. It differs in its construction from groin centering,as the former requires angle-ribs. There are two differentmethods of constructing groin-ceilings ; one by ribs fixedvertically and perpendicularly to the sides of each branch ;the other by angle-ribs and ceiling-joists, or straight pieces oftimber, running parallel to the axis of each branch, fixed tothe angle-ribs, and to other intermediate ribs, in verticalplanes, at right angles to the sides of each of the branches,and placed upon opposite piers, to shorten the bearing, inorder to make less scantlings for the ceiling-joists, and therebysave timber. But in whatever mode the groined-ceiling isconstructed, the surface must finish in the same manner. Itis evident, however, that the latter method by ceiling-joistswill require much less timber and workmanship than theformer, where so much stuff 1 is cut to waste, and so much timeemployed in making the angle-ribs.

Figure 1. The cradling of a groin ceiling, constructed withceiling-joists. Stout ribs are thrown across the angles andbetween the opposite piers ; the ceiling-joist being put onbelow and spiked upwards. In this, the rib, a lb c, of a trans-verse range is given, to find the others. Take any numberof points in the half arc alb ; draw lines parallel to g i, thebase of the rib of the body-range, to cut the diagonal df;from the points thus obtained in df, draw lines at right anglesto df, and make the corresponding perpendiculars equal to

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