The term cry pto-portiens was, however, applied, in course of time, to apartments similar to our galleries.
We find, in Pliny's description (if his house at Laurentian), that the e•ypto.portiens had windows on each side, looking towards the sea. timl upon his garden ; also other windows over these. When the weather was cold, they were shut on the side that sheltered them from the wind, but in warm and serene weather they were all set wide open.
CUB vru NE, or CUBATION, of a solid, the solid contents aceording to any co111111011 measure, as a solid inch, foot, yard, tSze.. which is called the measuring unit. The enbature is the 'same in respect to the contents of a solid, as the quadrature is in respect of a superficies.
There is one general rule that will apply to finding the cullation of nearly all the regular solids as entire bodies, or to their frustums or segments, viz. `• to four dines the area of the middle section add the area of each end ; multiply one-sixth of this sum (which is the mean area) by the distance between the ends, and the product will give the content of the solid." This rule has been applied to three particular cases, viz., in the prismoid, cask measuring, and the frustum of the hyperboloill ; it has also been demonstrated rather as a theoretical curiosity, than as a practical rule, at the end of Dr. II ntton's Mensuration, applied to solids generated from conic sections.
This rule, however, applies to solids in general, and mm prehends the whole of mensuration in its principle ; though such an extension has never been noticed by any writer on the subject, yet it cannot fail to be of the utmost use in assisting the memory ; for when particular rules are forgotten, this general one may he easily remembered. It will apply with accuracy to prisms, pyramids, prismoids, cones, colloids, cnneoids, spheres, spheroids, and to all their segments and frustums, cut by planes parallel to their axes.
Some may object, however, by saying it is not easy to come at the dimensions of the middle section ; hut in straight solids these will be very readily ascertained, by taking half the sum of the two ends; in complete spheres and spheroids, the middle dimension is absolutely given. In the hyperboloid
and its frustums, the dimensions of the middle section is much easier obtained than either the transverse or conjugate diameters, one of which, or both, it would otherwise be necessary to have. In the paraboloid and its frustums, the middle area is half the sum of the areas of the two ends. The reader who is desirous of seeing the application of' this general rule, may consult the article SoLm.
CUBE (from Kvdoc, texsera, die) a solid, bounded by six squares ; it is otherwise called a hexahedron, from its six sides. Its simple properties are, that its sides are all equal and at right angles with each other: it has also its opposite sides parallel to each other. The cube may be conceived to he generated by a square, figure along a right line, of equal length to the side of the square, and perpendicular to the plane. From its construction, it is evident that all sections of the solid, parallel to any side, are equal.
The envelope, rete, or net, may he thus constructed : Draw two lines, A it and A B, at a right angle with each other ; make A It equal to the side of one of the squares, and A it equal to four times A a, marking the points of division, E G 1; draw it c parallel to A D, and D c parallel to A B, parallel to which also draw E F, G II, T K, cutting B C at F. 0, K : produce E F and o u on both sides, to L and m on the one side, :111(1 N and o on the other, making E L, G M, F N, 11 0, each equal to E F, and join L m and N o; this will complete the envelope required.
fence the superfieies of a cube is found by multiplying the area of one of its sides by 6.
The solidity of a cube is found by multiplying the area of one of its squares by one of the lineal sides of a square. Ilonce if' one lineal side be 10, the solidity will he 1,000; and if 12, it will he 172.1 ; wherefore a cubic perch contains 1,000 cubic feet, and a cubic foot. of 1725 cubic, inches.
Cubes are to one another in the triplieate ratio of their lineal sides.