516 



CARPENTRY. 



Pyramid 



Cunius. 



A solid having any plane figure for its base, and 

 its sides triangles, meeting in a common point, is 



called a py ramid and this p int is call( r d the vet ; tex 



of the pyramid. A pyramid is denominated trian- 

 gular, square, pentagonal, hexagonal, &c. If the 

 base be a circle, the pyramid is called a cone. 



The axis of a cone is a straight line, extending 

 from the centre of the base to the vertex. 



Properties. 



All parallel sections of a pyramid are similar fi- 

 gures, except they be parallel to a plane within the 

 pyramid, passing through the vertex. 



All sections through the vertex within the pyra- 

 mid are triangles. 



All sections of a cone parallel to the plane, pass- 

 ing through the vertex without the cone, but not 

 parallel to the base, are ellipses. 



All sections of a cone parallel to the base are 

 circles. 



All sections of a cone parallel to a plane touching 

 the curved surface are parabolas. 



All sections of a cone parallel to a plane within the 

 eone, passing through the vertex, are hyperbolas. 



The frustum of a pyramid, or cone, is that which 

 is left by cutting away the part which contains the 

 vertex by a plane parallel to the base. 



An ungula of a pyramid, or cone, is that which is 

 left by cutting away the part which has the vertex, 

 by a plane not parallel to the base. 



A CUNIUS is a solid, the base of which is a rect- 

 angle, and the four sides joining the base plane sur- 

 faces ; two are triangles, and the other two parallel- 

 ograms. 



The cunioidj or cono-cunius, is a tapering solid, 

 such that the base is a circle, or ellipse, and the ver- 

 tex a straight line, parallel and equal to the diameter 

 of the base ; the curved surface such, that if a plane 

 be supposed to pass through the middle of the ver- 

 tex and the centre of the base, and a straight line be 

 continually applied while in motion to the circumfe- 

 rence and the vertex so as always to be parallel to 

 the plane, until the straight line so applied has gone 

 entirely round the circumference of the base. 



Properteis* 



All sections of the solid parallel to the plane, as 

 well as that in the plane, are triangles. 



All sections passing through -the vertex are paral- 

 lelograms. 



All sections parallel to the base are ellipses, except 

 one, which is a circle. 



If the vertex be in a plane, passing through the 

 centre of the base, perpendicular to the said base., 

 and the straight line drawn from the centre of the 

 base to the middle of the vertex be also perpendicu- 

 lar to the base, the cunioid is said to be right, but if 

 otherwise oblique. 



A PRISMOID is a solid, terminated by two dissimilar 

 rectangular ends, and the remaining surfaces joining 

 the ends planes, 



Properties. 



All parallel sections cutting entirely through the 

 sides, are quadrilaterals. 



All sections parallel to the bases have their corre- 

 sponding angles equal, but the corresponding sides 

 of these angles disproportional. 



A SPHERE is a solid, such that all lines drawn, or 

 conceived to be drawn, from a certain point within 

 the solid to the surface, are equal. 



Properly. 

 All the sections of a sphere are circles. 



A SPHEROID is a solid, formed by the revolution^ 

 of a semi-ellipse about one of the axes. 



If the spheroid be generated round the greater 

 axis, it is called an oblong or prolate spheroid. 



If the spheroid be generated round the lesser axis, 

 it is called an oblate spheroid. 



PROB. I. Given the altitude of any three points 

 above a plane, and the seats of the three points on 

 that plane, to find the intersection of another plane 

 passing through the points, with the plane which 

 contains the seats of the three points. 



Let A, B, C be the seats of the three given points. PLATE 

 Join any two points A and C ; draw AD and CE CXV. 

 each perpendicular to AC ; make AD equal to the s ' ! 2 ! 

 height of the point on A, CE equal to the height 

 of the point on C, and join ED ; produce ED and 

 CA to meet each other at I ; make CF on the line 

 CE equal to the height on B ; draw FG parallel to 

 CA, cutting ED at G ; and draw GH parallel to 

 EC, cutting AC at H ; join HB, and draw KIL 

 parallel to HB, then will KL be the intersection 

 required. 



In Fig. 1. the distance CF is greater than AD, 

 but less than CE ; this makes the point G fall be- 

 tween the points D and E, and the point H between 

 the points A and C. In Fig. 2. CF is greater than 

 CE ; this makes the point G fall on DE produced, 

 and the point H on AC produced. In Fig. 3. HG 

 is less than AD, which makes the point G fall be- 

 tween D and I, and the point H between A and I. 



These three diagrams are particularly useful in the 

 covering of solids, in the sections of cylinders and 

 prisms, and in the finding of the face moulds for the 

 hand rails of stairs, and in groins where the planes 

 of the angle ribs stand at oblique angles to the plan, 

 and where the section passes through three given 

 points. Fig. 1. and 2. are also useful in the sections 

 of solids, when the angle which the vertical side 

 makes with the inclined side is given. 



SOLID ANGLES* 



Of the Construction of Solid Angles, consisting of 

 any three Plane Angles. 



In these, besides the three plane angles, there are 

 also to be considered the three inclinations of the 



