SPA

437

SPA

o!° Ut an

Sjfrteach.

pi ets the width of an arch; the distance between theits Sn ’ ° r ^ le length of an imaginary line extending between§p’ n "t n g on either side ; the same as chord.

Oti e .VIOREL, an irregular triangular space, bounded ono, he> . by the outer or convex curve of an arch, and on the

inch broad, nailed to the joists at intervals of

each

otl

,w 0 by two straight lines touching the curve, and meet-

other at right angles.

%•

v >ti£r

It is commonly formed in, with a rectangular hood-

ie spandrel so formed being frequently enriched withor sculpture, sometimes with foliage, at others with

arc '" tec,:ure by an arch

toatr e f *i i .

q,| lu us, shields of arms, and various devices.for^ 16 tei '>n is further applied to any surface of a similarSPa*" ^ orme< l under similar circumstances.fi *edb ^ DREL BRACKETING, a cradling of bracketsari(j • et ween one or more curves, each in a vertical plane,t ^ le circumference of a circle, whose plane is liori-

Position I.— Given the plan of a square room to he«e C ; !o “ 0r vaulted with spandrel bracketing, and the verticalr °°oi ^ ie cove i whose plane passes through the axis of theto f}l? n d through the intersection of two adjoining walls;/so ej ® the figure of the springing on the walls, so that theby the brackets may be in the surface of a solid, formed%^ng the given section from Us place in the angle,In ax ‘ s °f ^ le room, until it makes a revolution.the c ' l ' e S‘ v en square inscribe a circle, which will representthe „ 1 ln g-hne, from any angle of the square draw a line toto j ts ntl ’c : apply the given section upon this line as a base,B* a ce at the angle. Take any number of points in thean< l draw lines perpendicular to the base to cut it.s ev 6p „, 16 centre of the square, and with the distance of thePoints of section, describe arcs cutting any side. S( lnare. From these points draw perpendiculars toPeritJi i I 10111 which make the several lengths of the per-tte ftiit' r ' 3 e< l ua l t0 those of the section; through the ex-

H>ri n J es draw a curve, which will be the figure of the£ e'ng.

1- Figure 1. — Let the given square be a b c b,

to K „ ; - -■

< ]ye

15 he its centre ; join e b ; apply the given section, b f,., as a base, so that when turned perpendicular round e ban' 6 sc l Ufll 'c, it will then be in its place. In the curve, f b,nu ‘nber of points, h, and draw the lines h i, perpen-' v itl 1 t j to E b, meeting it at the points i.

5Nts ‘r r; i < b> E i, describe arcs, i h, meeting a b in the

* to

si*-

On the centre, E,

radii

From the points k, draw the lines, k l, perpen-AB; and also from M, the point of contact,tl le j.' 1 ' Make the perpendiculars mn, hi, hi, &c. equal■ ^t've ° es 1 ^ : through the points b, l , l , l , &c., to n, draw'[ Il'ich will be half the springing line required. But^'SpQj 1 "tsy> are taken in the arc, mg, equidistantly, andc nts P be the upper ends of the brackets, it will be muchNh t j atlv cnient to draw the lines e p k. On the centre, e,. ^adii e h, describe the arcs k i; draw the perpen->'v e> * 1 and k l, and proceed as before, to complete theb^ets 8 U l e P°‘. nts 1 will give the spring of the foot of the; f In this example, the section here given is a quad-la e, ^ may be any geometrical curve, or any other

0 ttlb at fanc y may suggest, the same method extending

ol a I'ftnt^^ figure 2.—In this example, the section givent| Sq° a Clrc l e > Bie radius of which is half the diagonalti f ll! ds . a * e ’ Bterefore, the quadrant, one of whose radii isit t * le °B ler the half of the diagonal, will generate

s base , e ^ e- I* > s evident, that if a square be inscribed inI'ot, . j anc I cu t by planes through the sides of the square' 5 T

at right angles to the base, the sections will be semicircles ;therefore, upon any side, as a diameter, describe a semicircle,and it will give the springing required. The places orspringing-points of the brackets, may be found as before.

Proposition II.— Given the plan of an oblong room, tobe coved or vaulted with spandrel bracketing, the ceiling-linean ellipsis inscribed in the plan, and a section of the cove,through the diagonal perpendicular to the ceiling-line; to findthe spandrels of the bracketing, such that they may be the in-tersections made by the planes of the four walls, with thecurved surface of a solid, generated by the given section, mov-ing always perpendicular to the ceiling-line, which passesthrough the upper extremity of the curve of the generatingsection, and the lower extremity, always in a horizontalplane.

Figure 3.—Let the oblong, a b c d, be the plan of thewalls, and e f g h, the inscribed ellipsis ; e g being thegreater axis, and f h the lesser, parallel to the sides ofthe rectangle ; consequently, e, f, g, h, the points of contact.

In any quadrant, E H, take any number of equidistant points, i,or rather in a ratio, in which every succeeding two towards e,the extremity of the longer axis, will be nearer and nearertogether, by the property of the ellipsis. Draw the lines, i k,perpendicular to the curve, cutting the two sides, da, uc, inthe points k. Let N e O be the given section, n e its base,which is a prolongation of G E, the line drawn from D, per-pendicular to the curve, meeting it at r ; from the point e,on e n, make the distances e / respectively equal to thelines i k. Draw the lines l m perpendicular to n e, cuttingthe curve N O in the points m, and the lines kp perpendicularto A d and d C : make all the lines kp equal to the lines Im;and through the points p draw curves, which will be halfthe springing line on the two adjacent sides, and the points pwill be the places for the springing of the brackets. Thereader must here observe, that though the points l woulddescribe parallel curves on the horizontal plane, yet none ofthese curves would be ellipses.

Proposition III.— Given the plan of a rectangular roomto be coved with spandrel bracketing, supposing the surfaceof the solid with which the faces of the brackets are to coin-cide, to be an ellipsoid, the base of the spandrel solid, a sectionpassing through its axis, given the springing-line of one of the \ends, which is a section perpendicular to the fixed axis; tofind the springing-line upon the sides, so that they may be ofthe same height with that upon the ends, and also the curvesof the ribs.

Figure 4.—Let A b c d be the rectangular plan, andlet A F b be the section given, which is necessarily semicir-cular; upon a d, as a transverse axis, and with one-ha fof A b, as a semi-conjugate, describe the semi-ellipsis a g d,and it will be the springing required for the other two sides.Produce 10 and HP, the two axes of the inscribed ellipsis,in all the four directions. Through any point, a, in one ofthe angular points of the plane, describe an ellipsis, nkl m,whose two axes shall be on the lines HP and oi, and also inthe same ratio as those lines, the transverse axis being mk,and the conjugate N L ; this ellipsis would be the base of theellipsoid. Now, as every section passing through the shorteraxis of this ellipsoid, viz., all the vertical ellipses passingthrough e, have the same conjugate axis, it will be very easyto describe the ribs over any given seat, as this seat is thetransverse axis, and the radius of the greatest circle isthe semi-conjugate.

Two diagrams illustrative of the method of obtaining theprojection of angle-brackets are added.

Figure 1, a cove bracket ; bae the plan of the angle-lib ;the ordinates h i, are taken from the ordinates gf