Chap. VIII.

GEOMETRY.

777

Fig. 854.

To construct a Prism -Form on each side of the base

T its sides XY, of the length required, and also the topV of the size and form of the base, and then the wholemav be folded together to form the prism or figurerequired. In the figure shown at S, we have a simple rightprism, the faces being all perpendicular to the ends ; thedevelopments are consequently all rectangles which arejoined together at the edges and enclosed at each end.

V shows the plan of an hexagonal prism, and its mannerof uniting the sides Z to form a regular figure.

To draw an inclined prism, we commence by tracing theprofile parallel to its degree of inclination, and then de-scribing on its axis the plane which is perpendicular to it;afterwards by lengthening the lines which represent theedges, we may project the horizontal section, as is shown inthe development of right and oblique cylinders.

The Tetraedron. — The three triangles which surroundthat in the middle, and which are made to meet in a pointat the summit, form a figure with four faces. To formthis out of paper or card, it is only necessary to partly cutthrough the lines to enable it to fold.

A sphere may be circumscribed about a given tetraedron,as well as inscribed within it.

The sphere or globe may be considered the chief orprimary figure, and the standard of proportion to the tri-angular prism, the cylinder, and the cone : the discovery ofthe diagram of a sphere inclosed within a cylinder on atomb at Syracuse led Cicero to pronounce it as that ofArchimedes . In the triangle we might inscribe a circle,which would represent the base of the three secondaryfigures, and among the Saxon-sculptured ornaments wefind the globe encompassed by an equilateral triangle,which is readily effected by sinking the stone withoutthe circle, and then rounding ofT its edges to produce thehemisphere. Taquet, in his Theorems of Archimedes , hasadded “ Una tribus ratio est ” to the diagrams in which isrepresented the prism, the cylinder, and globe. Length,breadth, and depth constitute a geometrical solid, and thesethree dimensions belong to the parallelopipedon, the tetra-edron, and the sphere: the first of these figures is said tobe cubic, plinthic, and oblong; the second, equiangular,obtuse, or acute; and the third, right, oblate, and ovate; ofthese varieties there are many subdivisions. The sphere isthe most perfect of the whole, as it comprehends all othersin its centre, diameter, radii, and circumference.

To draw a Hexaedron or Cube y or to construct one, anexact square must be set out for a base; then on the sidesthe four squares RSTP, each equal to O, and attachedto P another square Q, which, when folded together, formsthe top. Care must be taken to make allowances for thejoining or uniting at the angles.

Cubes are variously represented: the top Dmay be shownof the same dimensions as the front C, or the double cubemay be drawn with its top, I K L M, double the face orfront GHIK; it may be shaded, as N, and the dotted linesE F omitted.

Mr. Peter Nicholson, in his Practical Treatise on Stone-cutting , says, “ Every stone bounded by six quadrilateralplanes or faces forms a solid, of which the surfaces terminateon eight points, every three surfaces in one point: everythree planes thus terminating is termed a solid angle ortrihedral The angles formed by the intersections of the ig '

faces with one another, or the three plane angles, are called sides of the trihedral, andthe angles of inclination, by way of distinction, simply angles. 'The three sides, aswell as the three angles, are each called a part, so that the whole trihedral consists of sixparts, and if any three of these be given, the remaining three can be found : therefore,if bodies constructed of stone, which are intended to have their solid angles to consist

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