208 



JOINERY. 



metnciU 



Joinery. Fig. 12. Shallow flutes, which may also be applied 

 Vt-> V 1 ' ' to columns, pilasters, and flat bands. 



Fig. 1 3. Style of a door or shutter, with part of the 

 panel, shewing the mouldings which are here termed 

 quirk, ogee, and bead. 



Fig. 1+. Style anil part of the panel of a door or shut- 

 ter, shewing the mouldings which are in this example 

 termed quirked ovolo and bead. 



Fig. 1 5. Section of a door style, with part of the pa- 

 nel, shewing the mouldings which are here termed 

 bolection mouldings. 



Figs. 16, 17, 18, and 19, are various forms of sections 

 for sash bars. 



Scribing and Mitreing. 



Scribing When two bodies are so fitted together that their 



nd mi- surfaces intersect or meet each other, they are in gene- 

 treirjg. ra l said to mitre or scribe. 



Two bodies are said to mitre together in a plane 

 passing through the common intersection of their sur- 

 faces. 



One body is said to scribe upon another, when the 

 two surfaces intersect each other, and when so much of 

 the one body is cut off to make way for the other 

 body entire. 



In finishing, whether the bodies are mitred or scribed, 

 the external appearance is the same. 



Theory of As the theory of the intersection of geometrical bo- 

 the interseo dies with one another has been omitted in the article 

 tion of geo- CARPENTRV, where it is absolutely necessary in theprac- 

 fi^ o f g ro ; ns an d arches, in order to make correct 

 work ; and as it is also essential in joinery, in mitreing, 

 and scribing, we shall make no apology for inserting it 

 in the present article. 



The bodies which we shall suppose to be joined to- 

 gether are prisms, cones, and conoids. 



Prisms include all solids which may be cut into 

 equal and similar sections by parallel planes, and which 

 may also be cut by parallel planes in some other di- 

 rection into parallelograms of the same length; and 

 consequently by this definition, not only triangular, rec- 

 tangular, and polygonal prisms are included, but also 

 cylinders, cylindroids, and such as may have parallel 

 sections, equal and similar parabolas, or equal and si- 

 milar hyperbolas. 



Parabolic and hyperbolic prisms are here supposed 

 to be generated in the following manner. Imagine the 

 plane of the figure to be, with its apex, along a straight 

 line perpendicular thereto, while its axis or double or- 

 dinate to the axis may describe a plane. 



In order to prevent repetitions, let it be understood, 

 that when two prisms intersect each other, that they 

 intersect at right angles. 



The method of ascertaining the construction of the 

 meeting of the surfaces of two different bodies, is to 

 suppose the position of the one body given in respect 

 to the other, and the position of both in respect to 

 a given plane, and the projection of the intersection of 

 the two surfaces to be made on that plane. 



For the purpose of projection, let us suppose that 

 besides the plane on which the projection is made, there 

 are two others at right angles, forming, with the plane 

 of projection, an internal solid angle. 



To render the practice of this easy, we shall suppose 

 that when the intersection of two prisms is requir- 

 ed, the ends are placed at right angles to the plane of 

 projection, and that the double ordinates of their gene- 

 rating figures are parallel thereto. 



Let us suppose, in the case of two prisms joining, 

 that the planes generated by the axis of the generating 



Joinery. 



Thtory of 

 the intersec- 

 tion of geo- 

 metrical 

 bodies. 



Problem. 



Example 1. 



PLATE 

 CCCXXXIII. 



Fig. ?0. 



figure of each prism, are the plane whose distances are 

 respectively x and y. 



Or, in the case of a prism joining with a conoid, that 

 the plane described by the axis of the generating figure 

 of the prism, and the plane passing through 1 he axis 

 or centre of the conoid at right angles therewith, and 

 also to the plane of projection, are thie two planes whose 

 distances are respectively x and y. 



PROB. To find the projection of the intersection of 

 two prisms, or of a prism and conoid. 



Suppose the plane of projection to pass through one 

 extremity of the axis of the generating figure, and let 

 a be equal to the axis of that figure, and consequently 

 equal to the distance of the most remote point of the 

 intersection of the two solids from the plane, of projec- 

 tion ; and let z be equal to the distance of any point in 

 the intersection from the said plane, x equal to the dis- 

 tance of that point from one of the vertical planes, and 

 y the distance of the -same point from the other vertical 

 plane. 



Find the equation of the one generating curve in 

 terms of x and z, also the equation of the other gene- 

 rating curve in terms of y and z. Find the value of z 

 or any equal power of s in each of these equations; put 

 these values equal to one another ; then the equation be- 

 tween x and y will determine the species of the curve. 



Ex. 1. Suppose two parabolic prisms to intersect each 

 other, so that the apex line of the one prism, and the 

 rectangle opposite the apex line of the other, may be in 

 the plane of projection. 



Fig. 20. Plate CCCXXXIII, Let CI be the apex line 

 of the one prism, and A'F the line described by the ex- 

 tremity of the axis which meets the double ordinate of 

 the other. Let A PDC be half of the generating para- 

 bola, of which its axis forms the line CI; and A'P'D'C' 

 half the generating section, of which the extremity of 

 the axis that meets the double ordinate forms the line 

 A'F. 



Draw BP parallel to CD ; make AC = a, CD = b, 

 AB = z, and BP =y. Draw B'P' parallel to C'D' ; 

 make C'A' = a, C'D' = c, C'B' = z, and B'P' = x ; 

 therefore B'A' will be = a z. 



Then, by the property of the parabola, 



In the section APDC, a : z : : 4 2 :y*; therefore 2= -JL; 



and in the section C'D'P'A', a: a z : : c s : a 2 ; there- 



ax 1 

 f OI ez = a ; 



Q- It Q 3^ 



Consequently ~ := a - ; whence we infer, that 



the curve FGHRI, which is the projection of the two 

 prismatic surfaces is an ellipse. 



If b = c, the projection will be a circle. 



Ex. 2. Suppose the generating figures of both prisms Example g. 

 to be parabolas, as before ; and that the rectangle de- 

 scribed by the double ordinate of each' is- on the plane 

 of projection, to find the projection of the intersection 

 of their surfaces. 



Let APDC, Fig. 21. Plate CCCXXXIII, be the gener- PIATE 

 ating figure of the one prism, and A'P'D'C' that of the cccxxxnr, 

 other. Make CA = a, CB = z, CD = b, and BP = x, f>s ' 81 * 

 also C'A' = c, C'B' = z, C' D' = d, and B'P' = y; 

 therefore BA = a 2, and B'A' = c x. 



From the property of the parabola we have from the 

 figure APDC, a : a z : : A 2 : x s ; whence z := a 

 ax 1 . 



-, also from the figure A'P'D'C, c : c z : : d* \f; 



6* 



whence = c 



cy* 

 ~~ 



therefore a - 





