Chap. VIII.

GEOMETRY.

781

quently it will also be perpendicular to all the joints of which their planes are parallel tothe axis.

The vertical projection of the intrados will be a curve similar to the curve of the rightsection of the intrados. The vertical projections of the coursing joints will be radiantstraight lines, intersecting the curved projection of the intrados. The vertical pro-jections of all the joints, which are in vertical planes parallel to the axis, will be straight,lines perpendicular to the ground line: the vertical projection of all the joints, horizontalplanes parallel to the ground line.

In practice we may mould a cylinder to the form of the opening, which is to be archedor vaulted over, and after its ends have been cut to suit the faces of the walls with whichit is to unite, we may make a correct model of this, and trace upon it the direction of eachcourse, showing its plan and situation; or we may cover it with plastic clay, of a thicknesscorrespondent with the first course of masonry, upon which we may define each particularjoint, and thus arrive at a thorough knowledge of all the planes and their projections.Models at all times serve to help us to the right knowledge of form, and when cor-rectly made, there is no difficulty in giving their representation upon paper, whatever maybe their situation with respect to the plane on which they are to be drawn. The same prin-ciples apply to cones and all other figures, whether right or oblique.

Cones are constructed bydescribing a circle of a radiusequal to its slant height, andthen dividing the circum-ference into six equal parts ;taking one of these parts, anduniting it together at theedge, the cone required isformed. When it is requiredto form a less acute cone, agreater portion of the circlemay be taken, or that portionwhich equals the circum-ference of the base.

To draw a cone, eitherperpendicular to the horizonor reclining, a circle, G, mustbe first described, of the ex-tent to be given to the base,as E F C D: G H must be set

Fig. 875.

out of the required height, and the perpendicular drawn,after which the sides H F, and II1), II C, and H E.

In France the epure made by the engineers and architectsfor the use of the workmen are of the size of the objects tobe formed ; these, corresponding with our working drawings,are set out with the greatest precision. The position ofevery part or its plane must be carefully studied before anyfigure can be worked. If in the cone X it were required totake out a portion, as that of TS in the cone Y, it is onlyrequisite to set out upon the card of which it is formed thedimension T S on its outer edge, and then lines drawn to thecentre, from which the curvature is struck, will represent theomitted part. By a similar method the conical covering of Fig 877a tower or building may have developed upon it all theapertures or ornaments which form its decoration ; which,when drawn upon the flat surface, will be in their trueposition after the card is folded together, if projected truly.

To cut out a piece of paper or card, so that it can befolded up into the form of an oblique pyramid : the positionof the point must be shown on the plan or horizontal pro-jection, which corresponds with the apex of the pyramid ;from this point must be described the arc of a circle, uponwhich are to be transferred the horizontal projections of theinclined arrises : then on the perpendicular raised from theplan to the apex is to be set the height of the pyramidabove the plane of projection; from a point so established,lines, as dotted, are to be drawn to their seat on the horizontalplane, which will show the real lengths of the edges of thepyramids: the small section shows the form at one-third Fig. 879.from the top.

Fig. 876.

Fig 878.